Bergman spaces for the bicomplex Vekua equation with bounded coefficients

Proposition 7. For $1 , the Cauchy integral operator <math>C_{\Gamma} : W^{1,1-\frac{1}{p}}(\Gamma) \to W^{1,p}(\Omega)$ is bounded. The proof is provided in Appendix A. In particular, the Borel-Pompeiu formula (19) is valid for functions $u \in W^{1,p}(\Omega)$ . Following [8], we introduce the bicomplex version of the Theodorescu and the Cauchy integral operators.

Proposition 9. The following statements hold. - (i) The operators in (20) can be written as $\mathbf{T}_{\Omega}W(z) = \mathbf{p}^{+}B_{\Omega}W^{+}(z) + \mathbf{p}^{-}A_{\Omega}W^{-}(z)$ and $\mathbf{C}_{\Gamma}\varphi(z) = \mathbf{p}^{+}\left(C_{\Gamma}(\varphi^{+})^{}(z)\right)^{} + \mathbf{p}^{-}C_{\Gamma}\varphi^{-}(z)$ , respectively. - (ii) For $1 \leq p \leq \infty$ , $\mathbf{T}_{\Omega} \in \mathcal{B}(L_p(\Omega; \mathbb{B}))$ and $\mathbf{T}_{\Omega}W \in \overline{\mathfrak{D}}_p(\Omega; \mathbb{B})$ with $\overline{\partial} \mathbf{T}_{\Omega}W = W$ for every $W \in L_p(\Omega; \mathbb{B})$ . - (iii) For $1 , <math>\mathbf{T}_{\Omega} \in \mathcal{B}(L_p(\Omega; \mathbb{B}), W^{1,p}(\Omega; \mathbb{B}))$ , and for $p = \infty$ , $\mathbf{T}_{\Omega} \in \mathcal{B}(L_{\infty}(\Omega), C^{0,\epsilon}(\Omega; \mathbb{B}))$ for all $0 < \epsilon < 1$ . - (iv) If $\Omega$ is of class $C^1$ , then $\mathbf{T}_{\Omega} \in \mathcal{K}(L_p(\Omega; \mathbb{B}))$ for all 1 . - (v) $\mathbf{C}_{\Gamma} \in \mathcal{B}\left(W^{1,1-\frac{1}{p}}(\Gamma,\mathbb{B}),W^{1,p}(\Omega;\mathbb{B})\right)$ for 1 . - (vi) Given $W \in W^{1,p}(\overline{\Omega}; \mathbb{B})$ , 1 , the bicomplex Borel-Pompeiu formula is valid: $$\mathbf{C}_{\Gamma}[\operatorname{tr}_{\Gamma} W](z) + \mathbf{T}_{\Omega} \left[ \overline{\partial} W \right](z) = W(z), \quad z \in \Omega.$$ (21) In particular, $\mathbf{T}\overline{\partial}V = V$ for all $V \in W_0^{1,p}(\Omega; \mathbb{B})$ . (vii) The adjoint of $\mathbf{T}_{\Omega}$ in the space $L_2(\Omega; \mathbb{B})$ is given by $$\mathbf{T}_{\Omega}^* W = -\mathbf{p}^+ A_{\Omega} W^+ - \mathbf{p}^- B_{\Omega} W^-. \tag{22}$$ Additionally, $-\partial \mathbf{T}_{\Omega}^W = W$ for all $W \in L_2(\Omega; \mathbb{B})$ , and $-\mathbf{T}_{\Omega}^\partial W = W$ for $W \in W_0^{1,2}(\Omega)$ .

Lemma 10. Suppose that $\Gamma$ is a Liapunov curve. Let $1 , <math>p' = \frac{p}{p-1}$ , and $V \in W^{1,p}(\Omega;\mathbb{B})$ which satisfies the following condition: <span id="page-8-0"></span> $$\int_{\Gamma} \psi(\zeta) V(\zeta) d\widehat{\zeta} = 0, \quad \forall \psi \in W^{1, 1 - \frac{1}{p'}}(\Gamma; \mathbb{B}).$$ (23) Then there exists $W_0 \in W_0^{1,p}(\Omega; \mathbb{B})$ and $G \in W^{1,p}(\Omega; \mathbb{B})$ $\mathbb{B}$ -analytic such that $$V = W_0 + G. (24)$$ Proof. The condition (23) can be rewritten as follows: <span id="page-8-1"></span> $$\int_{\Gamma} \psi(\zeta) \operatorname{tr}_{\Gamma} u(\zeta) d\zeta^* = 0, \quad \int_{\Gamma} \phi(\zeta) \operatorname{tr}_{\Gamma} v(\zeta) d\zeta = 0 \quad \forall \phi, \psi \in W^{1,1-\frac{1}{p'}}(\Gamma), \tag{25}$$ where $u = V^+$ and $v = V^-$ . We focus on the second integral. Condition (25) implies $\int_{\Gamma} \operatorname{tr}_{\Gamma} v(\zeta) \zeta^n d\zeta = 0$ for all $n \in \mathbb{N}_0$ . According to [22, Th. 4.5 of Sec. 2.4], $\operatorname{tr}_{\Gamma} u$ is the non-tangential limit of $g(z) = C_{\Gamma}[\operatorname{tr}_{\Gamma} v](z), z \in \Omega$ . By Proposition 7, $g \in W^{1,p}(\Omega)$ and then $\operatorname{tr}_{\Gamma} u = \operatorname{tr}_{\Gamma} g$ . Hence $w_0 = v - g \in W_0^{1,p}(\Omega)$ and $v = w_0 + g$ . Applying the same procedure to the integral $\int_{\Gamma} \psi^(\zeta) \operatorname{tr}_{\Gamma} u^(\zeta) d\zeta = 0$ , we obtain that $u = w_1 + h$ , where $w_1 \in W_0^{1,p}(\Omega)$ and $h \in W^{1,p}(\Omega)$ is anti-analytic. Thus, V = W + G, where $W = \mathbf{p}^+ w_1 + \mathbf{p}^- w_0 \in W^{1,p}(\Omega; \mathbb{B})$ and $G = \mathbf{p}^+ h + \mathbf{p}^- g \in W^{1,p}(\Omega; \mathbb{B})$ , which is $\mathbb{B}$ -analytic. $\blacksquare$

Proposition 13. The following statements hold. (i) The operator $\mathbf{S}_{\Omega}^{(a,b)}$ maps $\overline{\mathfrak{D}}_p(\Omega;\mathbb{B})$ into itself, and the following relation is valid: <span id="page-9-0"></span> $$\overline{\partial} \mathbf{S}_{\Omega}^{(a,b)} W = \left( \overline{\partial} - \mathbf{Q}_{(a,b)} \right) W, \quad \forall W \in \overline{\mathfrak{D}}_{p}(\Omega; \mathbb{B}). \tag{30}$$ - (ii) Let $1 . If <math>W \in L_p(\Omega; \mathbb{B})$ and $V \in L_p(\Omega; \mathbb{B})$ satisfy $\overline{\partial}W \mathbf{Q}_{(a,b)}W = V$ in the weak sense, then $W \in W_{loc}^{1,p}(\Omega; \mathbb{B})$ . - (iii) If $W \in L_{\infty}(\Omega; \mathbb{B})$ and $V \in L_{\infty}(\Omega; \mathbb{B})$ satisfy $\overline{\partial}W \mathbf{Q}_{(a,b)}W = V$ in the weak sense, then $W \in C_{locdisk}^{0,\epsilon}(\Omega; \mathbb{B})$ for all $0 < \epsilon < 1$ . Proof. Consider $W \in \overline{\mathfrak{D}}_p(\Omega; \mathbb{B})$ , and let $H = \mathbf{S}_{\Omega}^{(a,b)}W \in L_p(\Omega; \mathbb{B})$ . As $\mathbf{Q}_{(a,b)}W \in L_p(\Omega; \mathbb{B})$ , by Proposition 9(ii), we observe that $\mathbf{T}_{\Omega}\mathbf{Q}_{(a,b)}W \in \overline{\mathfrak{D}}_p(\Omega; \mathbb{B})$ , hence $H \in \overline{\mathfrak{D}}_p(\Omega; \mathbb{B})$ , and by Proposition 9(ii), $\overline{\partial}H = \overline{\partial}W - \mathbf{Q}_{(a,b)}W$ . This establishes (i). Now suppose that $1 . Since <math>W \in L_p(\Omega; \mathbb{B})$ is a solution of the non-homogenous Vekua equation with right-hand side $V \in L_p(\Omega; \mathbb{B})$ , $W \in \overline{\mathfrak{D}}_p(\Omega; \mathbb{B})$ . By (30), $H = \mathbf{S}_{\Omega}^{(a,b)}W$ is a weak solution of $\overline{\partial}H = V$ if and only if W is a weak solution of $(\overline{\partial} - \mathbf{Q}_{(a,b)})W = V$ . Note that equation $\overline{\partial}H = V$ and Remark 12 imply that $H = \mathbf{T}_{\Omega}V + G$ , with $G \in \mathcal{A}^p(\Omega; \mathbb{B})$ . Hence $H \in W_{loc}^{1,p}(\Omega; \mathbb{B})$ , by Proposition 9(iii). Thus, $W = H + \mathbf{T}_{\Omega}\mathbf{Q}_{(a,b)}W \in W_{loc}^{1,p}(\Omega)$ . When $p = \infty$ , by Proposition 9(iii), $\mathbf{T}_{\Omega}V$ and $\mathbf{T}_{\Omega}\mathbf{Q}_{(a,b)}W$ belongs to $C^{0,\epsilon}(\Omega; \mathbb{B})$ for all $0 < \epsilon < 1$ , from where we obtain (iii). Corollary 14 $H = \mathbf{S}_{\Omega}^{A,B}W \in \mathcal{A}^{p}(\Omega; \mathbb{B})$ iff $W \in \mathcal{A}^{p}_{(a,b)}(\Omega; \mathbb{B})$ . Hence $\mathcal{A}^{p}_{(a,b)}(\Omega; \mathbb{B}) \subset W^{1,p}_{loc}(\Omega; \mathbb{B})$ for $1 , and <math>\mathcal{A}^{\infty}_{(a,b)}(\Omega; \mathbb{B}) \subset C^{0,\epsilon}_{locdisk}(\Omega; \mathbb{B})$ for all $0 < \epsilon < 1$ . Consider the unbounded operator $\overline{\partial}: \operatorname{dom}(\overline{\partial}) \subset L_p(\Omega; \mathbb{B}) \to L_p(\Omega; \mathbb{B})$ with domain $\operatorname{dom}(\overline{\partial}) = \overline{\mathfrak{D}}_p(\Omega; \mathbb{B})$ , and let $\overline{\partial} - \mathbf{Q}_{(a,b)}: \overline{\mathfrak{D}}_p(\Omega; \mathbb{B}) \to L_p(\Omega; \mathbb{B})$ be the sum of $\overline{\partial}$ with the bounded operator $\mathbf{Q}_{(a,b)}$ .

Proposition 15. The operator $\overline{\partial} - \mathbf{Q}_{(a,b)} : \overline{\mathfrak{D}}_p(\Omega; \mathbb{B}) \to L_p(\Omega; \mathbb{B})$ is closed. Proof. The case when $a \equiv b \equiv 0$ is given by Remark 5. For the general case, consider $\{W_n\} \subset \overline{\mathfrak{D}}_p(\Omega; \mathbb{B})$ along with $W, V \in L_p(\Omega; \mathbb{B})$ such that $W_n \to W$ and $(\overline{\partial} - \mathbf{Q}_{(a,b)})W_n \to V$ in $L_p(\Omega; \mathbb{B})$ . Taking $H_n = \mathbf{S}_{\Omega}^{(a,b)}W_n$ and $H = \mathbf{S}_{\Omega}^{(a,b)}W$ , the continuity of $\mathbf{S}_{\Omega}^{(a,b)}$ and relation (30) imply that $H_n \to H$ and $\overline{\partial} H_n = V_n \to V$ in $L_p(\Omega; \mathbb{B})$ . Since $\overline{\partial}$ is closed, $H \in \overline{\mathfrak{D}}_p(\Omega; \mathbb{B})$ and $\partial H = V$ . By the equality $W = H + \mathbf{T}_{\Omega}\mathbf{Q}_{(a,b)}W \in \overline{\mathfrak{D}}_p(\Omega; \mathbb{B})$ and (30), the operator $\overline{\partial} - \mathbf{Q}_{(a,b)}$ is closed. $\blacksquare$

Theorem 16. The Vekua-Bergman space $\mathcal{A}^p_{(a,b)}(\Omega;\mathbb{B})$ is a Banach space, and for $1 is separable and reflexive. In particular, <math>\mathcal{A}^2_{(a,b)}(\Omega;\mathbb{B})$ is a separable Hilbert space. Proof. The $\mathcal{A}_{(a,b)}^p(\Omega;\mathbb{B})$ is a closed subspace of $L_p(\Omega;\mathbb{B})$ (and consequently, a Banach space) since it is the null space of the closed operator $\overline{\partial} - \mathbf{Q}_{(a,b)}$ . Since for $1 the space <math>L_p(\Omega;\mathbb{B})$ is separable and reflexive, $\mathcal{A}_{(a,b)}^p(\Omega;\mathbb{B})$ is also separable and reflexive (see [6], Propositions 3.21 and 3.25).

Proposition 17. Let $W \in \overline{\mathfrak{D}}_p(\Omega; \mathbb{B})$ be a solution of (31). For every $G \subseteq \Omega$ , there exists a constant $C_G > 0$ (which does not depend on W) such that <span id="page-10-1"></span> $$||W|_D||_{W^{1,p}(G;\mathbb{B})} \leqslant C_G \left(||V||_{L_p(\Omega;\mathbb{B})} + ||W||_{L_p(\Omega;\mathbb{B})}\right). \tag{32}$$ Proof. Let B be an open set with $G \subseteq B \subseteq \Omega$ and $\eta \in C_0^{\infty}(\Omega; \mathbb{B})$ a cut-off function satisfying $\eta \equiv 1$ in G and with supp $\eta \subset B$ . Hence $U = \eta W \in W_0^{1,p}(\Omega)$ (due to Proposition 13), and by Proposition 9(vi), $U = \mathbf{T}_{\Omega} \overline{\partial} U$ . Thus, $$U = \mathbf{T}_{\Omega} \left[ \eta \overline{\partial} W + W \overline{\partial} \eta \right] = \mathbf{T}_{\Omega} \left[ \eta \mathbf{Q}_{a,b} W \right] + \mathbf{T}_{\Omega} \left[ \eta V \right] + \mathbf{T}_{\Omega} \left[ W \overline{\partial} \eta \right].$$ In particular, the right-hand side equals W(z) for a.e. $z \in B$ . Hence $$||W|_{D}||_{W^{1,p}(G;\mathbb{B})} \leq ||\mathbf{T}_{\Omega}[\eta \mathbf{Q}_{a,b}W]||_{W^{1,p}(\Omega;\mathbb{B})} + ||\mathbf{T}_{\Omega}[\eta V]||_{W^{1,p}(\Omega;\mathbb{B})} + ||\mathbf{T}_{\Omega}[W\overline{\partial}\eta]||_{W^{1,p}(\Omega;\mathbb{B})}$$ $$\leq 2\operatorname{diam}(\Omega)[||\eta||_{L_{\infty}(\Omega;\mathbb{B})}(M_{1}||W||_{L_{p}(\Omega;\mathbb{B})} + ||V||_{L_{p}(\Omega;\mathbb{B})}) + ||\overline{\partial}\eta||_{L_{\infty}(\Omega;\mathbb{B})}||W||_{L_{p}(\Omega;\mathbb{B})}],$$ where $M_1 = \|\mathbf{Q}_{(a,b)}\|_{\mathcal{B}(L_p(\Omega;\mathbb{B}))}$ . Consequently, we obtain (32) with $C_G = 2 \operatorname{diam}(\Omega) \|\eta\|_{W^{1,\infty}(\Omega;\mathbb{B})} \max\{M_1,1\}$ .

Proposition 18. Let $W \in \overline{\mathfrak{D}}_p(\Omega; \mathbb{B})$ be a solution of (31) with $V \in L_q(\Omega; \mathbb{B})$ and $q \ge p$ , q > 2. The following statements hold. - (i) If p < 2, then $W \in C^{0,1-\frac{2}{r}}_{locdisk}(\Omega; \mathbb{B})$ for all $2 < r < \min\{p^*, q\}$ . - (ii) If $p \geqslant 2$ , then $W \in C^{0,1-\frac{2}{q}}_{locdisk}(\Omega; \mathbb{B})$ . - (iii) $\mathcal{A}^p_{(a,b)}(\Omega;\mathbb{B}) \subset C^{0,1-\frac{2}{r}}_{locdisk}(\Omega;\mathbb{B})$ for all $2 < r < p^*$ , if p < 2, and for all $p < r < \infty$ , if $p \geqslant 2$ . Proof. Take an arbitrary disk $D \in \Omega$ . By (30), we obtain the equality $$W|_D = G + \mathbf{T}_D \left( \mathbf{Q}_{(a,b)} W|_D + V|_D \right)$$ with $G \in \mathcal{A}^p(D; \mathbb{B})$ . Since $G \in C^{\infty}(D; \mathbb{B})$ , we focus on the integral $\mathbf{T}_D\left(\mathbf{Q}_{(a,b)}W|_D + V|_D\right)$ . - (i) When $1 , we know that <math>p^ > 2$ . Taking $2 < r < \min\{p^, q\}$ , by Remark 3 and Proposition 13(ii), $W|_D \in L_r(D; \mathbb{B})$ , and hence $\mathbf{Q}_{(a,b)}W|_D + V|_D \in L_r(D; \mathbb{B})$ . Due to Proposition 9(iii) and Remark 3, $\mathbf{T}_D\left(\mathbf{Q}_{(a,b)}W|_D + V|_D\right) \in C^{0,1-\frac{2}{r}}(\overline{D}; \mathbb{B})$ . By the arbitrariness of D, we obtain (i). - (ii) In the case $p \geq 2$ , by Remark 3 and Proposition 13, $\mathbf{Q}_{(a,b)}W|_D + V|_D \in L_q(D; \mathbb{D})$ . Again, by Proposition 9(iii) and Remark 3, $\mathbf{T}_D\left(\mathbf{Q}_{(a,b)}W|_D + V|_D\right) \in C^{0,1-\frac{2}{q}}(\overline{D}; \mathbb{B})$ . Due to the arbitrariness of D, we conclude (ii). - (iii) Taking $V \equiv 0$ , this follows from points (i) and (ii). We recall that for a bounded domain $G \subset \mathbb{C}$ , a function W belongs to the class $C^{1,\epsilon}(\overline{G};\mathbb{B})$ , $\epsilon \in (0,1]$ , if $W \in C^1(\overline{G};\mathbb{B})$ and its partial derivatives belong to $C^{0,\epsilon}(\overline{G};\mathbb{B})$ . We say that $W \in C^{1,\epsilon}_{locdisk}(\Omega;\mathbb{B})$ if $W|_D \in C^{1,\epsilon}(\overline{D};\mathbb{B})$ for any open disk $D \subseteq \Omega$ .

Proposition 19. If $a, b, V \in C^{0,\epsilon}_{locdisk}(\Omega; \mathbb{B})$ , then every solution $W \in \overline{\mathfrak{D}}_2(\Omega; \mathbb{B})$ of (31) belongs to $C^{1,\epsilon}_{locdisk}(\Omega; \mathbb{B})$ . In particular $\mathcal{A}^p_{(a,b)}(\Omega; \mathbb{B}) \subset C^{1,\epsilon}_{locdisk}(\Omega; \mathbb{B})$ . Proof. Let $D \in \Omega$ be a disk, and write $W|_D = G + \mathbf{T}_D(\mathbf{Q}_{(a,b)}W|_D + V|_D)$ with $G \in \mathcal{A}^p(D; \mathbb{B})$ . Since $G \in C^{\infty}(D)$ , it only remains to analyze $\mathbf{T}_D(\mathbf{Q}_{(a,b)}W|_D + V|_D)$ . Using that for any disk B with $D \in B \in \Omega$ , $V|_B \in L_q(B; \mathbb{D})$ , with $q = \frac{2}{1-\epsilon} > 2$ , Proposition 18(ii) implies that $W|_D \in C^{0,\epsilon}(\overline{D}; \mathbb{B})$ . Since $a|_D, b|_D \in C^{0,\epsilon}(\overline{D}; \mathbb{B})$ , by [19, Prop. 1.2.2], $\mathbf{Q}_{(a,b)}W|_D \in C^{0,\epsilon}(\overline{B}; \mathbb{B})$ . Consequently, $\mathbf{T}_D(\mathbf{Q}_{(a,b)}W|_D + V|_D) \in C^{1,\epsilon}(\overline{D}; \mathbb{B})$ [8, Prop. 10 (v)]. Therefore $W \in C^{1,\epsilon}_{locdisk}(\Omega; \mathbb{B})$ . Remark 20 For $a, b \in C^{0,\epsilon}_{locdisk}(\Omega; \mathbb{B})$ , let $\mathcal{V}_{(a,b)}(\Omega; \mathbb{B})$ be the class of classical solutions $W \in C^1(\Omega; \mathbb{B})$ of (26). Proposition 19 implies that $\mathcal{A}^p_{(a,b)}(\Omega; \mathbb{B}) = \mathcal{V}_{(a,b)}(\Omega; \mathbb{B}) \cap L_p(\Omega; \mathbb{B})$ . In this case, the theory of bicomplex classical solutions of (26) was developed in [8]. The space $\mathcal{V}_{(a,b)}(\Omega; \mathbb{B})$ is closed in the Fréchet space $C(\Omega; \mathbb{B})$ , with respect to the topology of the uniform convergence on compact subsets [8, Th. 13]. By Proposition 18, every $W \in \mathcal{A}^p_{(a,b)}(\Omega; \mathbb{B})$ is continuous in $\Omega$ and for every $z \in \Omega$ , the evaluation map $\mathcal{A}^2_{(a,b)}(\Omega; \mathbb{B}) \ni W \mapsto W(z) \in \mathbb{B}$ is well-defined. <span id="page-12-0"></span>Remark 21 Consider the case $a \equiv b \equiv 0$ . Let $K \subset \Omega$ be compact. Given $W \in \mathcal{A}^p(\Omega; \mathbb{B})$ , we have that $W^+ \in \overline{\mathcal{A}}^p(\Omega)$ and $W^- \in \mathcal{A}^p(\Omega)$ , the complex anti-analytic and analytic Bergman spaces, respectively. Then there exist constants $C_K^1$ , $C_K^2$ depending only on K such that $$\max_{z \in K} |W^{-}(z)| \leqslant C_K^1 ||W^{-}||_{L_p(\Omega)} \quad and \quad \max_{z \in K} |W^{+}(z)| \leqslant C_K^2 ||W^{+}||_{L_p(\Omega)}$$ (see [17, Ch. I]). Thus, $\max_{z \in K} |W(z)|_{\mathbb{B}} \leqslant C_K ||W||_{L_p(\Omega;\mathbb{B})}$ with $C_K = 2 \max\{C_K^1, C_K^2\}$ .

Theorem 22. For every $K \subset \Omega$ compact, there exists a constant $C_K > 0$ such that <span id="page-12-2"></span> $$\max_{z \in K} |W(z)|_{\mathbb{B}} \leqslant C_K ||W||_{L_p(\Omega)}, \quad \forall W \in \mathcal{A}^p_{(a,b)}(\Omega; \mathbb{B}).$$ (33) Proof. First, consider the case when $K = \overline{D}$ is a closed disk. Let B be an open disk with $\overline{D} \subset B \subseteq \Omega$ , and take $H = \mathbf{S}_B^{(a,b)}W|_B$ . Since $H \in \mathcal{A}^p(B;\mathbb{B})$ , by Remark 21, there exits a constant $C_{\overline{D}}^1$ satisfying $$\max_{z \in \overline{D}} |H(z)| \leqslant C_{\overline{D}}^{1} ||H||_{L_{p}(B;\mathbb{B})} = C_{\overline{D}}^{1} ||\mathbf{S}_{B}^{(a,b)} W|_{B} ||_{L_{p}(B;\mathbb{B})} \leqslant C_{\overline{D}}^{1} M_{1} ||W||_{L_{p}(\Omega;\mathbb{B})},$$ where $M_1 = \|\mathbf{S}_B^{(a,b)}\|_{\mathcal{B}(L_p(B;\mathbb{B}))}$ . Due to Remark 3 and the fact that p > 1, there exits r > 2 such that the embeddings <span id="page-12-1"></span> $$W^{1,p}(B; \mathbb{B}) \hookrightarrow L_r(B; \mathbb{B}) \quad \text{and} \quad W^{1,r}(B; \mathbb{B}) \hookrightarrow C^{0,1-\frac{2}{r}}(\overline{B}; \mathbb{B})$$ (34) are bounded, and let $\tilde{C}_B^r$ and $\hat{C}_B^r$ be their norms. On the other hand, by Proposition 19, there is a constant $\tilde{C}_B^p > 0$ , independent of W, such that $\|W\|_B \|_{W^{1,p}(B;\mathbb{B})} \leq \tilde{C}_B^p \|W\|_{L_p(\Omega;\mathbb{B})}$ . Since $W|_B \in W^{1,p}(B;\mathbb{B})$ , by Proposition 13 and (34), $V = \mathbf{T}_B \mathbf{Q}_{(a,b)} W|_B \in W^{1,r}(B;\mathbb{B})$ . Taking $M_2 = \|\mathbf{T}_B \mathbf{Q}_{(a,b)}\|_{\mathcal{B}(L_r(B;\mathbb{B}),W^{1,r}(B;\mathbb{B}))}$ , we obtain $$\max_{z \in \overline{D}} |V(z)|_{\mathbb{B}} \leq ||V||_{C^{0,1-\frac{2}{r}}(B;\mathbb{B})} \leq \hat{C}_B^r ||V||_{W^{1,r}(B;\mathbb{B})} \leq \hat{C}_B^r M_2 ||W||_{L_r(B;\mathbb{B})}$$ $$\leq \hat{C}_B^r M_2 \tilde{C}_B^r ||W||_{W^{1,p}(B;\mathbb{B})} \leq \hat{C}_B^r M_2 \tilde{C}_B^r \tilde{C}_B^p ||W||_{L_p(\Omega;\mathbb{B})}.$$ Taking $C_{\overline{D}} = \max\{C_{\overline{D}}^1 M_1, \hat{C}_B^r M_2 \tilde{C}_B^r \tilde{C}_B^p\}$ , we have that $\max_{z \in \overline{D}} |W(z)|_{\mathbb{B}} \leqslant C_{\overline{D}} \|W\|_{L_p(\Omega;\mathbb{B})}$ . For a general compact subset K, take $D_1, \ldots, D_N$ open disks with $K \subset \bigcup_{j=1}^N \overline{D}_j \subset \Omega$ . Hence inequality (33) is satisfied by taking $C_K = \max_{1 \leq i \leq N} C_{\overline{D}_j}$ .

Proposition 25. For any $A \in \mathbb{B}$ , the kernel $\mathscr{K}_{\Omega}^{(a,b)}(A;z,\zeta)$ admits the Fourier series <span id="page-14-0"></span> $$\mathscr{K}_{\Omega}^{(a,b)}(A;z,\zeta) = \sum_{n=0}^{\infty} \langle A, \Phi_n(\zeta) \rangle_{\mathbb{B}} \Phi_n(z). \tag{40}$$ The series converges in the variable z in the $L_2(\Omega; \mathbb{B})$ -norm and uniformly on compact subsets of $\Omega$ . Proof. The proof of equality (40) is the same as in [32, Remark 17]. The uniform convergence on compact subsets of $\Omega$ in the variable z follows from Theorem 22. Since $\mathcal{A}^2_{(a,b)}(\Omega;\mathbb{B})$ is a closed subspace of $L_2(\Omega;\mathbb{B})$ , there exists the bounded orthogonal projection of $L_2(\Omega;\mathbb{B})$ onto $\mathcal{A}^2_{(a,b)}(\Omega;\mathbb{B})$ [14, Ch. I, Th. 2.7], denoted as $\mathbf{P}^{(a,b)}_{\Omega}$ . We refer to it as the Vekua-Bergman projection. Similar to the case of the bicomplex main Vekua equation [32, Remark 18], the Vekua-Bergman projection can be written in terms of the kernel $\mathscr{K}^{(a,b)}_{\Omega}(A;z,\zeta)$ .

Proposition 26. The Vekua-Bergman projection can be written as $$\mathbf{P}_{\Omega}^{(a,b)}\Psi(z) = \iint_{\Omega} \mathcal{K}_{\Omega}^{(a,b)}(\Psi(\zeta), z, \zeta) dA_{\zeta}, \qquad \forall \Psi \in L_{2}(\Omega; \mathbb{B}). \tag{41}$$ Proof. The proof is the same as in [32, Remark 18]. In the case b = 0, the Vekua-Bergman space $\mathcal{A}^2_{(a,0)}(\Omega; \mathbb{B})$ becomes a $\mathbb{B}$ -module. Consequently, certain properties are analogous to those observed for the complex Bergman spaces, beginning with the behavior of the reproducing kernel itself.

Proposition 27. When b = 0, for every $A \in \mathbb{B}$ we have $\mathscr{K}_{\Omega}^{(a,0)}(A; z, \zeta) = AK_{\Omega}^{a}(z, \zeta)$ , where $K_{\Omega}^{a}(z, \zeta) = K_{\zeta}(z)$ and $K_{\zeta}$ is defined by (35). Thus, we have the reproducing property <span id="page-14-1"></span> $$W(z) = \iint_{\Omega} K_{\Omega}^{a}(z,\zeta)W(\zeta)dA_{\zeta} \qquad \forall W \in \mathcal{A}_{(a,0)}^{2}(\Omega;\mathbb{B}). \tag{42}$$ Proof. Let $K(z,\zeta) = K_{\zeta}(z)$ and $L(z,\zeta) = L_{\zeta}(z)$ as in (35). Using the equalities $Sc(\mathbf{j}W) = -\text{Vec }W$ , $Vec(\mathbf{j}W) = Sc W$ , and (8), we obtain $$\operatorname{Sc} L(z,\zeta) = \operatorname{Sc} L_{\zeta}(z) = \operatorname{Vec} (\mathbf{j}L_{\zeta}(z)) = \langle \mathbf{j}L_{\zeta}, L_{z} \rangle_{L_{2}(\Omega;\mathbb{B})} = -\langle L_{\zeta}, \mathbf{j}L_{z} \rangle_{L_{2}(\Omega;\mathbb{B})}$$ $$= -\left(\operatorname{Vec}(\mathbf{j}L_{z}(\zeta))\right)^{} = -\left(\operatorname{Sc} L(\zeta,z)\right)^{}.$$ Similarly, Sc $K(z,\zeta) = (\text{Vec } L(\zeta,z))^*$ . Using these equalities together with (36), we obtain $$L(z,\zeta) = \operatorname{Sc} L(z,\zeta) + \mathbf{j} \operatorname{Vec} L(z,\zeta) = - \left(\operatorname{Sc} L(\zeta,z)\right)^ + \mathbf{j} \left(\operatorname{Sc} K(\zeta,z)\right)$$ $$= - \operatorname{Vec} K(z,\zeta) + \mathbf{j} \operatorname{Sc} K(z,\zeta) = \mathbf{j} K(z,\zeta).$$ Substituting this equality into (38) and (39), we obtain that $K_{\Omega}^{(a,0)}(A;z,\zeta) = AK(z,\zeta)$ for every $A \in \mathbb{B}$ and the reproducing property (42). Remark 28 For the $\mathbb{B}$ -analytic Bergman space $\mathcal{A}^2(\Omega;\mathbb{B})$ , the Vekua-Bergman kernel is denoted by $\mathscr{K}_{\Omega}(z,\zeta)$ . Using the relation $W^{\pm}(z) = \left(\iint_{\Omega} \mathscr{K}_{\Omega}(z,\zeta)W(\zeta)dA_{\zeta}\right)^{\pm} = \iint_{\Omega} \mathscr{K}^{\pm}(z,\zeta)W^{\pm}(\zeta)dA_{\zeta}$ , we conclude that $\mathscr{K}_{\Omega}^{\pm}(z,\zeta)$ are the anti-analytic and the analytic complex Bergman kernels [17, Ch.I]. Remark 29 When $\mathcal{A}^2_{(a,b)}(\Omega;\mathbb{B})$ possesses a kernel $K^{(a,b)}_{\Omega}(z,\zeta)$ satisfying (42), it necessarily follows that $K(z,\zeta) = K^{(a,b)}_{\Omega}(z,\zeta)$ and $L(z,\zeta) = \mathbf{j}K^{(a,b)}_{\Omega}(z,\zeta)$ . Consequently, $\mathcal{K}^{(a,b)}_{\Omega}(A;z,\zeta) = AK^{(a,b)}_{\Omega}(z,\zeta)$ . Thus, the Vekua-Bergman projections takes the form $\mathbf{P}^{(a,b)}_{\Omega}\Psi(z) = \iint_{\Omega} K^{(a,b)}_{\Omega}(z,\zeta)\Psi(\zeta)dA_{\zeta}$ for all $\Psi \in L_2(\Omega;\mathbb{B})$ . For every $W \in \mathcal{A}^2_{(a,b)}(\Omega;\mathbb{B})$ , we have $\mathbf{j}W(z) = \mathbf{j}\iint_{\Omega} K^{(a,b)}_{\Omega}(z,\zeta)W(\zeta)dA_{\zeta} = \iint_{\Omega} K^{(a,b)}_{\Omega}(z,\zeta)\mathbf{j}W(\zeta)dA_{\zeta} = \mathbf{P}^{(a,b)}_{\Omega}[\mathbf{j}W](z)$ , hence $\mathbf{j}W \in \mathcal{A}^2_{(a,b)}(\Omega;\mathbb{B})$ and the Vekua-Bergman space is a $\mathbb{B}$ -module. Substituting $\mathbf{j}W$ into (26), we obtain the condition $b\overline{W} = 0$ for all $W \in \mathcal{A}^2_{(a,b)}(\Omega;\mathbb{B})$ . In particular, when b is a constant belonging to $\mathcal{R}(\mathbb{B})$ , such a kernel $K^{(a,b)}_{\Omega}(z,\zeta)$ cannot exists.

Proposition 30. The adjoint of the operator $\overline{\partial}$ is given by $(\overline{\partial})^* = -\partial : W_0^{1,2}(\Omega; \mathbb{B}) \to L_2(\Omega; \mathbb{B})$ . Proof. From the definition of the weak $\overline{\partial}$ -derivative, it is clear that $W_0^{1,2}(\Omega; \mathbb{B}) \subset \text{dom}(\overline{\partial})$ and $(\overline{\partial})^|_{W_0^{1,2}(\Omega;\mathbb{B})} = -\partial$ . Take $V \in \text{dom}(\overline{\partial})^*$ and $\Psi \in \overline{\mathfrak{D}}_2(\Omega;\mathbb{B})$ . Using the fact that this space is a $\mathbb{B}$ -module, the $\mathbb{B}$ -linearity of $\partial$ and $\overline{\partial}$ (Remark 5(ii)), and equalities (7) and (8), we obtain $$\iint_{\Omega} \overline{\partial} \Psi V^{\dagger} = \langle \overline{\partial} \Psi, V \rangle_{L_{2}(\Omega; \mathbb{B})} + \mathbf{j} \langle \overline{\partial} \Psi, \mathbf{j} V \rangle = \langle \Psi, (\overline{\partial})^{*} V \rangle_{L_{2}(\Omega; \mathbb{B})} - \mathbf{j} \langle \mathbf{j} \overline{\partial} \Psi, V \rangle_{L_{2}(\Omega; \mathbb{B})} = \langle \Psi, (\overline{\partial})^{} V \rangle_{L_{2}(\Omega; \mathbb{B})} - \mathbf{j} \langle \mathbf{j} \Psi, (\overline{\partial})^{} V \rangle_{L_{2}(\Omega; \mathbb{B})} = \iint_{\Omega} \Psi((\overline{\partial})^{*} V)^{\dagger}$$ Applying the involution † to both sides of the equality and using (17) we obtain <span id="page-15-0"></span> $$\iint_{\Omega} \partial \Psi V = \iint_{\Omega} \Psi(\overline{\partial})^* V \quad \text{for all } \Psi \in \overline{\mathfrak{D}}_2(\Omega; \mathbb{B}). \tag{43}$$ In particular, this is valid for $\Psi \in C_0^{\infty}(\Omega; \mathbb{B})$ , hence $-\partial V = (\partial)^ V$ . By Proposition 9(vii), $V = \mathbf{T}_{\Omega}^(\partial)^ V + G$ , with $G \in L_2(\Omega; \mathbb{B})$ a $\mathbb{B}$ -anti-analytic function. Since $\mathbf{T}_{\Omega}^(\overline{\partial})^ \subset (\overline{\partial}\mathbf{T}_{\Omega})^ = \mathbf{I}$ (by Proposition 9(ii) and [29, Prop.1.7]), V = V + G, and G = 0. Thus, $V = \mathbf{T}_{\Omega}^(\partial)^ V \in W^{1,2}(\Omega; \mathbb{B})$ . Finally, given $\psi \in W^{1,\frac{1}{2}}(\Gamma; \mathbb{B})$ , let $\Phi \in W^{1,2}(\Omega; \mathbb{B})$ such that $\mathrm{tr}_{\Gamma} \Phi = \psi$ . By the Green identities (18) and (43), we have $$\frac{-1}{2\pi \mathbf{j}} \int_{\Gamma} \psi(\zeta)^{\dagger} \operatorname{tr}_{\Gamma} V(\zeta) d\zeta^{\dagger} = \iint_{\Omega} \partial \Phi^{\dagger} V + \iint_{\Omega} \Phi^{\dagger} \partial V = 0,$$ which implies $\int_{\Gamma} \psi(\zeta) \operatorname{tr}_{\Gamma} V^{\dagger}(\zeta) d\zeta = 0$ for all $\psi \in W^{1,\frac{1}{2}}(\Omega; \mathbb{B})$ . Lemma 10 implies that $V = W_0 + G^{\dagger}$ , with $W_0 \in W_0^{1,2}(\Omega; \mathbb{B})$ and $G \in \mathcal{A}^2(\Omega; \mathbb{B}) \cap W^{1,2}(\Omega; \mathbb{B})$ . Hence, for every $\Phi \in W^{1,2}(\Omega; \mathbb{B})$ , from (43), we derive $$\iint_{\Omega} \boldsymbol{\partial} \Phi(W_0 + G^{\dagger}) = -\iint_{\Omega} \Phi \boldsymbol{\partial} W_0, \quad \text{ and thus, } \iint_{\Omega} G^{\dagger} \boldsymbol{\partial} \Phi = 0.$$ By choosing $\Phi = -\mathbf{T}_{\Omega}^G$ , we deduce that $\iint_{\Omega} G^{\dagger}G = 0$ , and consequently G = 0. Therefore $V = W_0 \in W_0^{1,2}(\Omega; \mathbb{B})$ and $(\partial)^ = -\partial$ . <span id="page-16-0"></span>Remark 31 Let $\mathbf{A}$ : dom( $\mathbf{A}$ ) $\subset \mathcal{H} \to \mathcal{H}$ be a densely defined operator in a Hilbert space $\mathcal{H}$ . When $\mathcal{A}$ is closed, according to [29, Prop. 16(ii) and Th. 1.8 (iii)], ker $\mathbf{A}$ is a closed subspace, $\mathbf{A}^{**} = \mathbf{A}$ and ker( $\mathbf{A}$ ) = (Im $\mathbf{A}$ ) $^{\perp}$ . Consequently, this yields the decomposition $$\mathcal{H} = \ker \mathbf{A} \oplus \overline{\operatorname{Im}(\mathbf{A}^*)}. \tag{44}$$ Corollary 32 (Bicomplex Hodge decomposition) The following equality holds <span id="page-16-1"></span> $$L_2(\Omega; \mathbb{B}) = \mathcal{A}^2(\Omega; \mathbb{B}) \oplus \partial W_0^{1,2}(\Omega; \mathbb{B}). \tag{45}$$ Proof. Since $\overline{\partial}$ is closed, by Remark 31, we obtain the decomposition $L_2(\Omega; \mathbb{B}) = \ker(\overline{\partial}) \oplus \overline{\operatorname{Im}(\overline{\partial})} = \mathcal{A}^2(\Omega; \mathbb{B}) \oplus \overline{\partial W_0^{1,2}(\Omega; \mathbb{B})}$ . It remains to show that $\partial W_0^{1,2}(\Omega; \mathbb{B})$ is closed. By the Maximum principle, if $W \in W_0^{1,2}(\Omega; \mathbb{B})$ and $\overline{\partial}W = 0$ , then $W \equiv 0$ . Furthermore, according to Prop. 9(vii), the bounded operator $\mathbf{T}_{\Omega}$ is an extension of the inverse of $-\partial$ . Thus, 0 is a regular point of $-\partial$ and, consequently, its range is closed [29, Prop. 2.1 (iv)]. Therefore (45) follows.

Proposition 33. The adjoint of $\overline{\partial} - \mathbf{Q}_{(a,b)}$ is $-\partial - a^{\dagger} - b^ \mathbf{C}_{\mathbb{B}} : W_0^{1,2}(\Omega; \mathbb{B}) \to L_2(\Omega; \mathbb{B})$ , where $b^ := (\operatorname{Sc} b)^ + \mathbf{j}(\operatorname{Vec} b)$ . Proof. Since $\overline{\partial} - \mathbf{Q}_{(a,b)}$ is the difference between $\overline{\partial}$ and the bounded operator $\mathbf{Q}_{(a,b)}$ , its adjoint is given by $(\overline{\partial} - \mathbf{Q}_{(a,b)})^ = (\overline{\partial})^ - (a\mathbf{I})^ - (b\mathbf{C}_{\mathbb{B}})$ with domain $\mathrm{dom}(\overline{\partial})^ = W_0^{1,2}(\Omega; \mathbb{B})$ (see [29, Prop. 1.6]). A direct computation shows that $(a\mathbf{I})^ = a^{\dagger}\mathbf{I}$ and $(b\mathbf{C}_{\mathbb{B}})^ = b^\mathbf{C}_{\mathbb{B}}$ , with $b^ = (\mathrm{Sc}\,b)^ + \mathbf{j}(\mathrm{Vec}\,b)^*$ .

Proposition 34. The following orthogonal decomposition holds: $$L_2(\Omega; \mathbb{B}) = \mathcal{A}^2_{(a,b)}(\Omega; \mathbb{B}) \oplus \overline{(\boldsymbol{\partial} + a^{\dagger} + b^* \mathbf{C}_{\mathbb{B}}) W_0^{1,2}(\Omega; \mathbb{B})}^{L_2(\Omega; \mathbb{B})}.$$ (46) Proof. It follows from Propositions 30 and 33, and Remark 31. ■ <span id="page-16-3"></span>Remark 35 By (28), $\|\mathbf{T}_{\Omega}\mathbf{Q}_{(a,b)}\|_{\mathcal{B}(L_{2}(\Omega;\mathbb{B}))} \leq 2\sqrt{2}\operatorname{diam}(\Omega)\max\{\|a\|_{L_{\infty}(\Omega;\mathbb{B})},\|b\|_{L_{\infty}(\Omega;\mathbb{B})}\}$ . In particular, when $\max\{\|a\|_{L_{\infty}(\Omega;\mathbb{B})},\|b\|_{L_{\infty}(\Omega;\mathbb{B})}\}<\frac{1}{2\sqrt{2}\operatorname{diam}(\Omega)}$ , $\mathbf{S}_{\Omega}^{(a,b)}\in\mathcal{G}\left(L_{2}(\Omega;\mathbb{B})\right)$ [2, Th. 10.23]. By the same reasoning the operator $\tilde{\mathcal{S}}_{\Omega}^{(a,b)}:=\mathbf{I}-\mathbf{T}^{}\mathbf{Q}_{(a,b)}^{}$ is invertible. <span id="page-17-0"></span>Remark 36 Let $D_{(a,b)}$ and $D_{(a,b)}$ denote the solution spaces of the Dirichlet problems $(\overline{\partial} - \mathbf{Q}_{(a,b)})W = 0$ and $(\partial + \mathbf{Q}_{(a,b)}^)W = 0$ , respectively, where $W \in W_0^{1,2}(\Omega; \mathbb{B})$ . Thus, $D_{(a,b)} \subset \ker \mathbf{S}_{\Omega}^{(a,b)}$ and $D_{(a,b)}^* \subset \ker \tilde{\mathbf{S}}_{\Omega}^{(a,b)}$ . In particular, both spaces have finite dimensions. We establish the inclusion $D_{(a,b)} \subset \ker \mathbf{S}_{\Omega}^{(a,b)}$ (the proof of $D_{(a,b)}^* \subset \ker \widetilde{\mathcal{S}}_{\Omega}^{(a,b)}$ is analogous). Let $W \in D_{(a,b)}$ . By the Borel-Pompeiu formula we have $W = \mathbf{T}\overline{\partial}W = \mathbf{T}\mathbf{Q}_{(a,b)}W$ , hence $\mathbf{S}_{\Omega}^{(a,b)}W = 0$ , and consequently, $W \in \ker \mathbf{S}_{\Omega}^{(a,b)}$ . Theorem 37 (Hodge decomposition) If $\max\{\|a\|_{L_{\infty}(\Omega;\mathbb{B})}, \|b\|_{L_{\infty}(\Omega;\mathbb{B})}\} < \frac{1}{2\sqrt{2}\operatorname{diam}(\Omega)}$ , then the following decomposition holds <span id="page-17-1"></span> $$L_2(\Omega; \mathbb{B}) = \mathcal{A}^2_{(a,b)}(\Omega; \mathbb{B}) \oplus \left(\partial + a^{\dagger} + b^* \mathbf{C}_{\mathbb{B}}\right) W_0^{1,2}(\Omega; \mathbb{B}). \tag{47}$$ Proof. According to Remark 35, $\tilde{S}_{\Omega}^{(a,b)} \in \mathcal{G}(L_2(\Omega;\mathbb{B}))$ . As stated in Remark 36, 0 is not an eigenvalue of the operator $\partial + \mathbf{Q}_{(a,b)}^ : W_0^{1,2}(\Omega;\mathbb{B}) \to L_2(\Omega;\mathbb{B})$ . Moreover, the inverse of $-(\partial + \mathbf{Q}_{(a,b)}^)$ is $\mathbf{R} = \left(\tilde{S}_{\Omega}^{(a,b)}\right)^{-1} \mathbf{T}_{\Omega}$ . Indeed, since $\tilde{S}_{\Omega}^{(a,b)}(\mathfrak{D}_2(\Omega;\mathbb{B})) \subset \mathfrak{D}_2(\Omega;\mathbb{B})$ and $-\partial \tilde{S}_{\Omega}^{(a,b)}W = -(\partial + \mathbf{Q}_{(a,b)}^)W$ for all $W \in \mathfrak{D}_2(\Omega;\mathbb{B})$ , we have that $\partial V = -(\partial + \mathbf{Q}_{(a,b)}^*)\left(\tilde{S}_{\Omega}^{(a,b)}\right)^{-1}V$ for all $V \in \mathfrak{D}_2(\Omega;\mathbb{B})$ . Thus, when $W \in W_0^{1,2}(\Omega;\mathbb{B})$ , using Proposition 9(vii), we obtain $$-\mathbf{R}\left(\boldsymbol{\partial} + \mathbf{Q}_{(a,b)}^\right)W = \left(\tilde{\mathcal{S}}_{\Omega}^{(a,b)}\right)^{-1} \left(-\mathbf{T}_{\Omega}^\boldsymbol{\partial}W - \mathbf{T}_{\Omega}^\mathbf{Q}_{(a,b)}^W\right) = \left(\tilde{\mathcal{S}}_{\Omega}^{(a,b)}\right)^{-1}\tilde{\mathcal{S}}_{\Omega}^{(a,b)}W = W$$ Again, by Proposition 9(vii), given $V \in L_2(\Omega; \mathbb{B})$ , we have $$-(\boldsymbol{\partial} + \mathbf{Q}_{(a,b)}^)\mathbf{R}V = -(\boldsymbol{\partial} + \mathbf{Q}_{(a,b)}^)\left(\tilde{\mathcal{S}}_{\Omega}^{(a,b)}\right)^{-1}\mathbf{T}_{\Omega}^V = -\boldsymbol{\partial}\mathbf{T}^V = V.$$ Therefore $-(\partial + \mathbf{Q}^*_{(a,b)})$ has a bounded inverse and 0 is a regular value, implying that its range is closed [29, Prop. 2.1 (iv)], yielding (47). Remark 38 In [8], the bicomplex exponential of $W \in \mathbb{B}$ is defined as $e^W := \mathbf{p}^+ e^{W^+} + \mathbf{p}^- e^{W^-}$ . For every $W, V \in \mathbb{B}$ , $e^{V+W} = e^W e^V$ , and, in particular, $e^W \in \mathcal{R}(\mathbb{B})$ with $(e^W)^{-1} = e^{-W}$ (for the proof of these facts, see [8, Prop. 4]). Note that $e^{\widehat{z}}$ is $\mathbb{B}$ -analytic. A direct computation shows that for every $V \in W^{1,2}(\Omega; \mathbb{B})$ , $\overline{\partial} e^V = \overline{\partial} V \cdot e^V$ . Thus, $\Phi_a := e^{\mathbf{T}_{\Omega}a}$ is a particular solution of $(\overline{\partial} - a) W = 0$ . Since $a \in L_{\infty}(\Omega; \mathbb{B})$ , $\Phi_a \in C^{0,\epsilon}(\overline{\Omega}; \mathbb{B}) \cap W^{1,2}(\Omega; \mathbb{B})$ for all $0 < \epsilon < 1$ . A right-inverse operator for $\overline{\partial} - a$ is given by $\mathbf{R}_{a,\Omega} V := \Phi_a \mathbf{T}_{\Omega} \Phi_{-a} V$ . Furthermore, every $W \in \mathcal{A}^2_{(a,0)}(\Omega; \mathbb{B})$ can be written as $W = \Psi \Phi_a$ , with $\Psi = W \cdot \Phi_{-a} \in \mathcal{A}^2(\Omega; \mathbb{B})$ . This is a version of the similarity principle for bicomplex pseudoanalytic functions [8, Th. 14]. Actually, the operator $\mathbf{R}_a := \Phi_a \mathbf{T}_{\Omega} \Phi_{-a}$ is not only a bounded right-inverse for $(\overline{\partial} - a)$ : $W_0^{1,2}(\Omega; \mathbb{B}) \to L_2(\Omega; \mathbb{B})$ but also a left-inverse: $$\mathbf{R}_{a} \left( \overline{\partial} - a \right) W = \Phi_{a} \mathbf{T}_{\Omega} \Phi_{-a} \left( \overline{\partial} W - aW \right) = \Phi_{a} \mathbf{T}_{\Omega} \left( \Phi_{-a} \overline{\partial} W - a\Phi_{-a} W \right)$$ $$= \Phi_{a} \mathbf{T}_{\Omega} \overline{\partial} (\Phi_{-a} W) = \Phi_{a} \Phi_{-a} W = W.$$ Hence 0 is a regular value for $(\overline{\partial} - a)$ . Similarly, the operator $\tilde{\mathbf{R}}_a := \Psi_{-a^{\dagger}} \mathbf{T}_{\Omega}^ \Psi_{a^{\dagger}}$ , where $\Psi_a := e^{-\mathbf{T}_{\Omega}^ a}$ , is the bounded inverse of $-(\partial + a^{\dagger}) : W_0^{1,2}(\Omega; \mathbb{B}) \to L_2(\Omega; \mathbb{B})$ . Thus, 0 is a regular value, and $(\partial + a^{\dagger}) W_0^{1,2}(\Omega; \mathbb{B})$ is closed. Consequently, the Hodge decomposition (47) is still valid for all $a \in L_{\infty}(\Omega; \mathbb{B})$ when $b \equiv 0$ .

Proposition 39. For all $W \in \mathcal{A}_{f}^{p}(\Omega; \mathbb{B})$ , the function $U = \frac{u}{f}$ , where $u = \operatorname{Sc} W$ , is a weak solution in $W_{loc}^{1,p}(\Omega)$ of the conductivity equation <span id="page-18-3"></span> $$\operatorname{div}\left(f^{2}\nabla U\right) = 0 \quad in \ \Omega. \tag{53}$$ Similarly, V = fv, where v = Vec W is a weak solution in $W_{loc}^{1,p}(\Omega)$ of <span id="page-18-4"></span> $$\operatorname{div}\left(\frac{1}{f^2}\nabla V\right) = 0 \quad in \ \Omega. \tag{54}$$ Proof. Since $u \in W^{1,p}_{loc}(\Omega; \mathbb{B})$ and $\frac{1}{f} \in W^{1,\infty}(\Omega)$ , we have $U = \frac{u}{f} \in W^{1,p}_{loc}(\Omega)$ . Given $\varphi \in C_0^{\infty}(\Omega)$ , equations (51) and (52) yield $$\begin{split} \iint_{\Omega} f^2 \nabla U \cdot \nabla V \varphi &= \iint_{\Omega} f \left\{ f \frac{\partial}{\partial x} \left( \frac{u}{f} \right) \frac{\partial \varphi}{\partial x} + f \frac{\partial}{\partial y} \left( \frac{u}{f} \right) \frac{\partial \varphi}{\partial y} \right\} \\ &= \iint_{\Omega} f \left\{ \frac{1}{f} \frac{\partial (vf)}{\partial y} \frac{\partial \varphi}{\partial x} - \frac{1}{f} \frac{\partial (vf)}{\partial y} \frac{\partial \varphi}{\partial y} \right\} = - \iint_{\Omega} vf \left\{ \frac{\partial^2 \varphi}{\partial y \partial x} - \frac{\partial^2 \varphi}{\partial x \partial y} \right\} = 0. \end{split}$$ Thus, U is a weak solution of (53). The proof of (54) is analogous. $\blacksquare$ We will require the following lemma.

Lemma 40. Let $G \subset \mathbb{C}$ be a bounded domain, and $\sigma \in W^{1,\infty}(G)$ such that $\frac{1}{\sigma} \in L_{\infty}(G)$ . Denote the space of weak solutions in $W^{1,p}(G)$ of the conductivity equation $\operatorname{div} \sigma \nabla u = 0$ by $$\operatorname{Sol}_{\sigma}^{p}(G) := \left\{ u \in W^{1,p}(G) \middle| \forall \varphi \in C_{0}^{\infty}(G) \int_{G} \sigma \nabla u \cdot \nabla \varphi = 0 \right\}. \tag{55}$$ The following statements hold: - (i) $\operatorname{Sol}_{\sigma}^{p}(G)$ is a closed subspace of $W^{1,p}(G)$ . - (ii) $\operatorname{Sol}_{\sigma}^{p}(G) \subset W_{loc}^{2,q}(G)$ , where $p \leqslant q \leqslant p^{*}$ if $1 , and <math>p \leqslant q < \infty$ if $p \geqslant 2$ . - (iii) $\operatorname{Sol}_{\sigma}^{p}(G) \subset C_{locdisk}^{1,1-\frac{2}{r}}(G)$ , where $2 < r < p^{*}$ if 1 , and <math>r > 2 if $p \geqslant 2$ . In particular, $\operatorname{Sol}_{\sigma}^{p}(G) \subset C^{1}(G)$ . - (iv) For every $G' \subseteq G$ , there exists $C_{G'} > 0$ such that $$||u||_{C^1(G')} \le C_{G'}||u||_{W^{1,p}(G)}, \quad \forall u \in \operatorname{Sol}_{\sigma}^p(G).$$ (56) The proof is given in Appendix A. Corollary 41 If $f \in C^1(\Omega) \cap W^{1,\infty}(\Omega)$ , then $\mathcal{A}_f^p(\Omega; \mathbb{B}) \subset C^1(\Omega; \mathbb{B})$ , that is, every weak solution of the main Vekua equation is a classical solution. Proof. Let $W \in \mathcal{A}_f^p(\Omega; \mathbb{B})$ and $u = \operatorname{Sc} W$ , $v = \operatorname{Vec} W$ . Take $G \subseteq \Omega$ . By Propositions 18 and 39, $U = \frac{u}{f} \in \operatorname{Sol}_{f^2}^p(G)$ and $V = fv \in \operatorname{Sol}_{\frac{1}{f^2}}^p(G)$ . It follows from Lemma 40(iii) that $U, V \in C^1(G)$ . Since $f \in C^1(\Omega)$ , $u = fU, v = \frac{V}{f} \in C^1(G)$ . Due to the arbitrariness of G, we conclude that $u, v \in C^1(\Omega)$ . The following result generalizes a well-known property for classical solutions of (48) when $f \in C^2(\Omega)$ (see [8, 25]). Corollary 42 Suppose that $f \in W^{2,\infty}(\Omega)$ . For every $W \in \mathcal{A}_f^p(\Omega; \mathbb{B})$ , $u = \operatorname{Sc} W \in W^{2,p}_{loc}(\Omega)$ satisfies the Schrödinger equation <span id="page-19-1"></span> $$-\triangle u + q_f u = 0, \quad in \ \Omega, \tag{57}$$ with potential $q_f := \frac{\triangle f}{f}$ , and $v = \operatorname{Vec} W \in W^{2,p}_{loc}(\Omega)$ satisfies de Darboux-associated equation <span id="page-19-2"></span> $$-\triangle u + q_{\frac{1}{f}}u = 0, \quad in \ \Omega, \tag{58}$$ where $q_f = f \triangle \left(\frac{1}{f}\right)$ is the Darboux-transformed potential of $q_f$ . Proof. Since $U = \frac{u}{f} \in \operatorname{Sol}_{f^2}^p(\Omega)$ and $V = fv \in \operatorname{Sol}_{\frac{1}{f^2}}^p(\Omega)$ , by Lemma 40, $U, V \in W_{loc}^{2,p}(\Omega)$ and satisfy (53) and (54) a.e. in $\Omega$ , respectively. Since $f \in W^{2,\infty}(\Omega)$ , $u, v \in W_{loc}^{2,p}(\Omega)$ . A direct computation shows that equations (53) and (54) are equivalent to (57) and (58), respectively. Denote $$\widehat{\operatorname{Sol}}_{f}^{p}(\Omega) = \left\{ u \in W^{1,p}(\Omega) \,\middle|\, U = \frac{u}{f} \in \operatorname{Sol}_{f^{2}}^{p}(\Omega) \right\}, \,\, \widehat{\operatorname{Sol}}_{\frac{1}{f}}^{p}(\Omega) = \left\{ v \in W^{1,2}(\Omega) \,\middle|\, V = fv \in \operatorname{Sol}_{\frac{1}{f^{2}}}^{p}(\Omega) \right\}. \tag{59}$$ Since the product by a proper conductivity is a bounded operation in $W^{1,p}(\Omega)$ , $\widehat{\operatorname{Sol}}_f^p(\Omega)$ and $\widehat{\operatorname{Sol}}_f^p(\Omega)$ are closed subspaces of $W^{1,p}(\Omega)$ . In the case when $f \in C^2(\Omega)$ and $\Omega$ is a simply connected domain, in [8, Sec. 3.4] it was shown that for every classical solution $u \in C^2(\Omega)$ of (57), a solution $v \in C^2(\Omega)$ of (58) such that $W = u + \mathbf{j}v$ is a classical solution of (48) (sometimes called a metaharmonic conjugate of u [25, Sec. 33]) is given by $$v = \frac{1}{f}\overline{A}\left[\mathbf{j}f^2\overline{\partial}\left(\frac{u}{f}\right)\right] + \frac{c}{f} \tag{60}$$ where c is an arbitrary complex constant and the operator $\overline{A}$ is defined as $$\overline{A}\Phi(z) = 2\int_{\gamma} (\operatorname{Sc}\Phi dx + \operatorname{Vec}\Phi dy) = 2\operatorname{Sc}\int_{\gamma} \overline{\Phi(\zeta)} d\widehat{\zeta}.$$ (61) The integral is taken over any curve that joins a fixed point $z_0 \in \Omega$ with z. For the integral to be well-defined, the condition <span id="page-20-0"></span> $$\frac{\partial}{\partial y} \operatorname{Sc} \Phi - \frac{\partial}{\partial x} \operatorname{Vec} \Phi = 0 \quad \text{in } \Omega, \tag{62}$$ is required. In such a case, $\overline{\partial A}W = W$ . We generalize this construction to weak solutions for the case when $f \in W^{1,\infty}(\Omega)$ and $\Omega$ is a star-shaped domain with respect to z = 0. In this case, taking $z_0 = 0$ and $\gamma$ as the line segment joining 0 with z, we introduce the operator $$I_f u(z) := \overline{A} \left[ \mathbf{j} f^2 \overline{\partial} \left( \frac{u}{f} \right) \right] = \int_0^1 f^2(tz) \left( x U_y(tz) - y U_x(tz) \right) dt, \text{ with } U = \frac{u}{f}.$$

Proposition 43. $I_f \in \mathcal{B}(\widehat{\operatorname{Sol}}_f^p(\Omega), W^{1,p}(\Omega))$ and $\nabla I_f u = f^2(U_x, -U_y)$ for all $u \in \widehat{\operatorname{Sol}}_f^p(\Omega)$ . Proof. Let $u \in \widehat{\mathrm{Sol}}_f^p(\Omega)$ and $U = \frac{u}{f}$ . By Lemma (40)(iii), $U \in C^1(\Omega)$ and hence $I_f u(z)$ exists for every $z \in \Omega$ . Note that $$\iint_{\Omega} |I_f u(z)|^p dA_z \leqslant \iint_{\Omega} \left[ \int_0^1 |f(tz)|^{2p} |x U_y(tz) - y U_x(tz)|^p dt \right] dA_z$$ $$\leqslant \operatorname{diam}(\Omega)^p ||f||_{L_{\infty}(\Omega)}^{2p} \int_0^1 \left[ \iint_{\Omega} |\nabla U(tz)|^p dA_z \right] dt.$$ Consider the family of operators $\{B_t\}_{0 \leq t \leq 1}$ given by $B_t u(z) := |\nabla U(tz)|$ for $u \in \widehat{Sol}_f^p(\Omega)$ . For t > 0, $t\Omega \subset \Omega$ (because $\Omega$ is radial) and changing variables we have $$||B_t u||_{L_p(\Omega)}^2 = \iint_{\Omega} |\nabla U(tz)|^p = \frac{1}{t^2} \iint_{t\Omega} |\nabla U(\zeta)|^p dA_{\zeta} \leqslant \frac{1}{t^2} ||\nabla U||_{L_p(\Omega)}^p \leqslant \frac{1}{t^2} M_f^p ||u||_{W^{1,p}(\Omega)}^p,$$ with $M_f = \|f^{-1}\|_{W^{1,\infty}(\Omega)}$ . For t = 0, $B_0u(z) = |\nabla U(0)|$ (a constant function). By Lemma 40(iv), there is a constant $C_0$ , independent of U, satisfying $|\nabla U(0)| \leq C_0 \|U\|_{W^{1,p}(\Omega)} \leq C_0 M_f \|u\|_{W^{1,p}(\Omega)}$ . Thus, $\{B_t\}_{0 \leq t \leq 1} \subset \mathcal{B}(\widehat{\operatorname{Sol}}_f^p(\Omega), L_p(\Omega))$ . Let $\rho_0 = \operatorname{dist}(0,\Gamma)$ and $\rho_1 = \sup_{z \in \Omega} |z|$ . Hence $\rho_0 \mathbb{D} \subset \Omega \subset \rho_1 \mathbb{D}$ . From the previous Let $\rho_0 = \operatorname{dist}(0,\Gamma)$ and $\rho_1 = \sup_{z \in \Omega} |z|$ . Hence $\rho_0 \mathbb{D} \subset \Omega \subset \rho_1 \mathbb{D}$ . From the previous calculation, $\|B_t u\|_{L_p(\Omega)}^p \leqslant \left(\frac{\rho_1}{\rho_0}\right)^2 M_f^p \|u\|_{W^{1,p}(\Omega)}$ for $\frac{\rho_0}{\rho_1} \leqslant t \leqslant 1$ . On the other hand, for $0 < t < \frac{\rho_0}{\rho_1}$ , we have $t\Omega \subset t\rho_1 \mathbb{D}$ and $$\operatorname{Area}(t\Omega) \geqslant \operatorname{Area}(t\rho_0 \mathbb{D}) = \left(\frac{\rho_0}{\rho_1}\right)^2 t^2 \rho_1^2 \operatorname{Area}(\mathbb{D}) = \left(\frac{\rho_0}{\rho_1}\right)^2 \operatorname{Area}(t\rho_1 \mathbb{D}),$$ from where we obtain $$||B_{t}u||_{L_{p}(\Omega)}^{p} = \frac{\operatorname{Area}(\Omega)}{\operatorname{Area}(t\Omega)} \iint_{t\Omega} |\nabla U(\zeta)|^{p} dA_{\zeta}$$ $$\leq \operatorname{Area}(\Omega) \left(\frac{\rho_{1}}{\rho_{0}}\right)^{2} \frac{1}{\operatorname{Area}(t\rho_{1}\mathbb{D})} \iint_{t\rho_{1}\mathbb{D}} |\nabla U(\zeta)|^{p} dA_{\zeta}$$ $$\leq \operatorname{Area}(\Omega) \left(\frac{\rho_{1}}{\rho_{0}}\right)^{2} \sup_{0 < r < \rho_{0}} \frac{1}{\operatorname{Area}(r\mathbb{D})} \iint_{r\mathbb{D}} |\nabla U(\zeta)|^{p} dA_{\zeta}.$$ The supremum $H_{|\nabla U|^p}(0) = \sup_{0 < r < \rho_0} \frac{1}{\operatorname{Area}(r\mathbb{D})} \iint_{r\mathbb{D}} |\nabla U(\zeta)|^p dA_{\zeta}$ is the Hardy-Littlewood maximal function of $|\nabla U|^p$ evaluated at z = 0. Since $\nabla U$ is continous $\Omega$ , z = 0 is a Lebesgue point and $H_{|\nabla U|^p}(0) < \infty$ (actually, $|\nabla U(0)|^p = \lim_{r \to 0^+} \frac{1}{\operatorname{Area}(r\mathbb{D})} \iint_{r\mathbb{D}} |\nabla U(\zeta)|^p dA_{\zeta}$ , see [20, Sec. 3.4]). Finally, for all $u \in \operatorname{Sol}_f(\Omega)$ we obtain the inequality $$\sup_{0\leqslant t\leqslant 1}\|B_t u\|_{L_p(\Omega)}^p\leqslant \left\{C_0^p M_f^p\operatorname{Area}(\Omega)\|u\|_{W^{1,p}(\Omega)}^p,\left(\frac{\rho_1}{\rho_0}\right)^2\operatorname{Area}(\Omega)H_{|\nabla U|^p}(0),\left(\frac{\rho_1}{\rho_0}\right)^2M_f^p\|u\|_{W^{1,p}(\Omega)}^p\right\}<\infty.$$ Since $\widehat{\mathrm{Sol}}_f^p(\Omega)$ is a Banach space, by the uniform boundedness principle [20, Th. 5.13], $M_1 = \sup_{0 \le t \le 1} \|B_t\|_{\mathcal{B}(\widehat{\mathrm{Sol}}_f^p(\Omega), L_p(\Omega))} < \infty$ . Thus, $$||I_f u||_{L_p(\Omega)}^p \leqslant \operatorname{diam}(\Omega)^p ||f||_{L_\infty(\Omega)}^{2p} \int_0^1 ||B_t u||_{L_p(\Omega)}^p dt \leqslant \operatorname{diam}(\Omega)^p ||f||_{L_\infty(\Omega)}^{2p} M_1^p ||u||_{W^{1,p}(\Omega)}^p.$$ For the differentiability, a direct computation shows that $\phi = f^2 U_y$ and $\psi = -f^2 U_x$ satisfy condition (62). Taking $\varphi \in C_0^{\infty}(\Omega)$ and $G \subseteq \Omega$ with supp $\varphi \subset G$ , we have $$\iint_{\Omega} I_f u(z) \varphi_x(z) dA_z = \iint_{G} \left[ \int_{0}^{1} (x \phi(tz) + y \psi(tz)) dt \right] \varphi_x(z) dA_z$$ $$= \int_{0}^{1} \left[ \iint_{G} (x \phi(tz) + y \psi(tz)) \varphi_x(z) dA_z \right] dt.$$ For every t > 0, we obtain $$\iint_{G} (x\phi(tz) + y\psi(tz))\varphi_{x}(z)dA_{z} = -\iint_{G} (\phi(tz) + tx\phi_{x}(tz) + ty\psi_{x}(tz))\varphi(z)dA_{z}$$ $$= -\iint_{G} (\phi(tz) + tx\phi_{x}(tz) + ty\phi_{y}(tz))\varphi(z)dA_{z},$$ where the last equality is by (62). Note that $\frac{d}{dt}(t\phi(tz)) = \phi(tz) + tx\phi_x(tz) + ty\phi_y(tz) \in L_p(\Omega)$ for every t > 0. Given $\epsilon > 0$ , we get $$\int_{\epsilon}^{1} \left[ \iint_{G} (\phi(tz) + tx\phi_{x}(tz) + ty\phi_{y}(tz))\varphi(z)dA_{z} \right] dt = \iint_{G} \left[ \int_{\epsilon}^{1} \frac{d}{dt} (t\phi(tz))dt \right] \varphi(z)dA_{z}$$ $$= \iint_{G} (\phi(z) - \epsilon\phi(\epsilon z))\varphi(z)dA_{z}.$$ Since $\phi \in C(\overline{G})$ (because $U_x, U_y \in C(\overline{G})$ ) and $f \in C(\overline{G})$ by Remark 3), the integrant is uniformly bounded in z and $\varepsilon$ . Thus, by dominated convergence $$\iint_{\Omega} I_f u(z) \varphi_x(z) dA_z = -\lim_{\epsilon \to 0^+} \iint_{G} \left( \phi(z) - \epsilon \phi(\epsilon z) \right) \varphi(z) dA_z = -\iint_{G} \phi(z) \varphi(z) dA_z.$$ Therefore $\frac{\partial}{\partial x}I_f u = \phi = f^2 U_x$ . Similarly, $\frac{\partial}{\partial y}I_f u = \psi = -f^2 U_y$ and hence $\|\nabla I_f u\|_{L_p(\Omega)} \le \|f\|_{L_\infty(\Omega)}^2 M_f \|u\|_{W^{1,p}(\Omega)}$ . Consequently, $I_f \in \mathcal{B}(\widehat{\operatorname{Sol}}_f^p(\Omega), W^{1,p}(\Omega))$ .

Theorem 44. Let $\Omega \subset \mathbb{C}$ be a bounded star-shaped domain with respect to z = 0. Given $u \in \widehat{\mathrm{Sol}}_f^p(\Omega)$ , $$v = \frac{1}{f} I_f u \in \widehat{\mathrm{Sol}}_{\frac{1}{f}}^p(\Omega) \quad and \quad W = u + \mathbf{j} v \in \mathcal{A}_f^p(\Omega; \mathbb{B}) \cap W^{1,p}(\Omega; \mathbb{B}). \tag{63}$$ If $v_1 \in W^{1,p}(\Omega)$ is another function such that $W = u + \mathbf{j}v_2 \in \mathcal{A}_f^p(\Omega; \mathbb{B}) \cap W^{1,2}(\Omega; \mathbb{B})$ , then $v_2 = \frac{1}{f}I_fu + \frac{c}{f}$ , where $c \in \mathbb{C}$ . Proof. Let $u \in \operatorname{Sol}_f^p(\Omega)$ . By Proposition 43, $v = \frac{1}{f}I_fu \in W^{1,p}(\Omega)$ and $\nabla I_fu = f^2(U_y, -U_x)$ . Given $\varphi \in C_0^{\infty}(\Omega)$ , we get $$\iint_{\Omega} \frac{1}{f^2} \nabla (fv) \cdot \nabla \varphi = \iint_{\Omega} \frac{1}{f^2} \nabla I_f u \cdot \nabla \varphi = \iint_{\Omega} (U_y \varphi_x - U_x \varphi_y) = -\iint_{\Omega} U(\varphi_{xy} - \varphi_{yx}) = 0.$$ Thus, $v \in \widehat{\mathrm{Sol}}_{\frac{1}{f}}^p(\Omega)$ . Taking $W = u + \mathbf{j}v = fU + \mathbf{j}\frac{1}{f}I_fu \in W^{1,p}(\Omega;\mathbb{B})$ , we obtain $$\left(\overline{\partial} - \frac{\overline{\partial} f}{f} \mathbf{C}_{\mathbb{B}}\right) W = f \overline{\partial} U + \frac{\mathbf{j}}{f} \overline{\partial} I_f u = f \overline{\partial} U - f \overline{\partial} U = 0,$$ and $W \in \mathcal{A}_f^p(\Omega; \mathbb{B})$ . If $v_2 \in W^{1,p}(\Omega)$ is another function such that $W_2 = u + \mathbf{j}v_2 \in \mathcal{A}_f^p(\Omega; \mathbb{B})$ , then $\mathbf{j}(v - v_2) = W - W_2 \in \mathcal{A}_f^p(\Omega; \mathbb{B})$ , and $0 = \left(\overline{\partial} - \frac{\overline{\partial} f}{f} \mathbf{C}_{\mathbb{B}}\right) W = \frac{\mathbf{j}}{f} \overline{\partial} f(v - v_2)$ . This is equivalent to $\nabla f(v_2 - v_2) = 0$ . Therefore $v_2 = v + \frac{c}{f}$ for some constant $c \in \mathbb{C}$ . A similar result is valid for the operator associated with $\frac{1}{f}$ and v.

Theorem 45. Let $\Omega \subset \mathbb{C}$ be a star-shaped bounded domain with respect to z=0. The operator $I_{\frac{1}{\ell}} \in \mathcal{B}(\widehat{\operatorname{Sol}}_{\frac{1}{\ell}}^p(\Omega), W^{1,p}(\Omega))$ and for any $v \in \widehat{\operatorname{Sol}}_{\frac{1}{\ell}}^p(\Omega)$ $$u = -fI_{\frac{1}{f}}u \in \widehat{\mathrm{Sol}}_{f}^{p}(\Omega) \quad and \quad W = u + \mathbf{j}v \in \mathcal{A}_{f}^{p}(\Omega; \mathbb{B}) \cap W^{1,p}(\Omega; \mathbb{B}). \tag{64}$$ If $u_2 \in W^{1,p}(\Omega)$ is another function such that $W = u + \mathbf{j}v \in \mathcal{A}_f^p(\Omega; \mathbb{B})$ , then $u_2 = -fI_{\frac{1}{f}}v + fc$ , where $c \in \mathbb{C}$ . Remark 46 Suppose that p=2 and $\Omega$ is a Lipschitz domain. Since $\frac{1}{f} \in L_{\infty}(\Omega)$ , Eq. (53) is strongly elliptic in $\Omega$ [28, p. 119]. Let $\varphi \in W^{1,\frac{1}{2}}(\Gamma)$ . There exists a unique $u \in \widehat{Sol}_f^2(\Omega)$ such that $\operatorname{tr}_{\Gamma}\left(\frac{u}{f}\right) = \varphi$ , and moreover, there is a constant $C_{\Omega,f} > 0$ , which does not depend on $\varphi$ such that $\|u\|_{W^{1,2}(\Omega)} \leq C_{\Omega,f}\|\varphi\|_{W^{1,\frac{1}{2}}(\Gamma)}$ (see [18, Sec. 6.2] and [28, Ch. 4]). Thus, it is possible to define a composition of operators as in the following diagram $$W^{1,\frac{1}{2}}(\Gamma) \xrightarrow{u} \widehat{\operatorname{Sol}}_{f}^{2}(\Omega) \xrightarrow{\frac{1}{f}I_{f}u} \widehat{\operatorname{Sol}}_{\frac{1}{f}}^{2}(\Omega) \xrightarrow{\operatorname{tr}_{\Gamma}} W^{1,\frac{1}{2}}(\Gamma).$$ Then the operator $\mathcal{H}_f: W^{1,\frac{1}{2}}(\Gamma) \to W^{1,\frac{1}{2}}(\Gamma)$ given by $\mathcal{H}_f \varphi := \operatorname{tr}_{\Gamma} \left( \frac{1}{f} I_f u \right)$ is a type of "Hilbert transform", in the sense that $I + \mathbf{j} \mathcal{H}_f : W^{1,\frac{1}{2}}(\Gamma) \to W^{1,\frac{1}{2}}(\Gamma; \mathbb{B})$ provides the trace of a function $W \in \mathcal{A}_f^2(\Omega; \mathbb{B}) \cap W^{1,2}(\Omega; \mathbb{B})$ such that $\operatorname{tr}_{\Gamma} \operatorname{Sc} W = \varphi$ . Finally, Proposition 44 allows us to extend the so-called Runge property to the main Vekua equation, for the case p = 2.

Theorem 47. Let $\Omega_1 \subseteq \Omega$ be a bounded Lipschitz star-shaped domain with respect to z = 0 such that $\Omega \setminus \overline{\Omega_1}$ is connected. Suppose that f is real-valued. Then for every $\varepsilon > 0$ and $W_1 \in \mathcal{A}_f^2(\Omega_1; \mathbb{B}) \cap W^{1,2}(\Omega_1; \mathbb{B})$ , there exists $W_2 \in \mathcal{A}_f^2(\Omega; \mathbb{B}) \cap W^{1,2}(\Omega; \mathbb{B})$ satisfying $$\|W_1 - W_2\|_{\Omega_1}\|_{W^{1,2}(\Omega_1:\mathbb{B})} < \varepsilon.$$ (65) Proof. Let $W_1 \in \mathcal{A}_f^2(\Omega_1; \mathbb{B}) \cap W^{1,2}(\Omega_1; \mathbb{B})$ and $\varepsilon > 0$ . By Proposition 44, $W = u + \frac{\mathbf{i}}{f}I_fu + \frac{c}{f}\mathbf{j}$ , where $u = \operatorname{Sc} W \in \widehat{\operatorname{Sol}}_f^2(\Omega_1)$ and $c \in \mathbb{C}$ . Since f is a real-valued proper conductivity, Eq. (53) satisfies the Runge property [26, Th. 3.9]. Hence, for $U_1 = \frac{u}{f} \in \operatorname{Sol}_{f^2}^2(\Omega_1)$ , there exists $U_2 \in \operatorname{Sol}_{f^2}^2(\Omega)$ satisfying $||U_1 - U_2|_{\Omega_1}||_{W^{1,2}(\Omega_1)} < \frac{\varepsilon}{C_f}$ . where $C_f = ||f||_{W^{1,\infty}(\Omega)} \left(1 + \left\|\frac{1}{f}I_f\right\|_{\mathcal{B}(\operatorname{Sol}_f^2(\Omega_1,),W^{1,2}(\Omega_1))}\right)$ . By Proposition 44, $W_2 = fU_1 + \frac{\mathbf{i}}{f}I_f(fU_2) + \frac{c}{f}\mathbf{j} \in \mathcal{A}_f^2(\Omega;\mathbb{B}) \cap W^{1,2}(\Omega;\mathbb{B})$ . Note that $I_f[fU_2|_{\Omega_1}] = I_f[fU_2]|_{\Omega_1}$ . Thus, $$||W_{1} - W_{2}|_{\Omega_{1}}||_{W^{1,2}(\Omega_{1};\mathbb{B})} \leq ||fU_{1} - fU_{2}||_{W^{1,2}(\Omega_{1})} + \left\| \frac{1}{f} I_{f}[fU_{1}] - \frac{1}{f} I_{f}[fU_{2}|_{\Omega_{1}}] \right\|_{W^{1,2}(\Omega_{1})}$$ $$\leq \left( ||f||_{W^{1,\infty}(\Omega)} + \left\| \frac{1}{f} I_{f} \right\|_{\mathcal{B}(\operatorname{Sol}_{f}^{2}(\Omega_{1}),W^{1,2}(\Omega_{1}))} ||f||_{W^{1,\infty}(\Omega)} \right) ||U_{1} - U_{2}||_{W^{1,2}(\Omega_{1})}$$ $$< \varepsilon.$$

Definition 8. Let $1 . The <math>\mathbb{B}$ -Theodorescu and the $\mathbb{B}$ -Cauchy operators are defined by <span id="page-7-0"></span> $$\mathbf{T}_{\Omega}W(z) := \frac{1}{\pi} \iint_{\Omega} \frac{W(\zeta)}{\widehat{z} - \widehat{\zeta}} dA_{\zeta}, \quad and \quad \mathbf{C}_{\Gamma}\varphi(z) := \frac{1}{2\pi \mathbf{j}} \int_{\Gamma} \frac{\varphi(\zeta)}{\widehat{\zeta} - \widehat{z}} d\widehat{\zeta}, \tag{20}$$ for $W \in L_p(\Omega; \mathbb{B})$ and $\psi \in L_p(\Gamma, \mathbb{B})$ , respectively.

Definition 11. The Vekua-Bergman space is the class of all weak solutions $W \in L_p(\Omega; \mathbb{B})$ of Eq. (26), denoted by $\mathcal{A}^p_{(a,b)}(\Omega; \mathbb{B})$ . <span id="page-9-1"></span>Remark 12 For the case $a \equiv b \equiv 0$ , we obtain the $\mathbb{B}$ -analytic Bergman space $\mathcal{A}^p(\Omega; \mathbb{B}) := \mathcal{A}^p_{(0,0)}(\Omega; \mathbb{B})$ . By Remark $\mathcal{A}^p(\Omega; \mathbb{B}) = \{W \in C^1(\Omega; \mathbb{B}) \cap L_p(\Omega; \mathbb{B}) \mid W \text{ is } \mathbb{B}\text{-analytic}\}.$ Denote $\mathbf{Q}_{(a,b)}W := aW + b\overline{W}$ . Since $a, b \in L_{\infty}(\Omega; \mathbb{B})$ , then $\mathbf{Q}_{(a,b)} \in \mathcal{B}(L_p(\Omega))$ . By (9), we have the inequality <span id="page-9-3"></span> $$\|\mathbf{Q}_{(a,b)}\|_{\mathcal{B}(L_p(\Omega;\mathbb{B}))} \leqslant \sqrt{2} \max\{\|a\|_{L_{\infty}(\Omega;\mathbb{B})}, \|b\|_{L_{\infty}(\Omega;\mathbb{B})}\}.$$ $$(28)$$ Following [11, 16], we introduce the operator $$\mathbf{S}_{\Omega}^{(a,b)}W := W - \mathbf{T}_{\Omega}[\mathbf{Q}_{(a,b)}W]. \tag{29}$$ Again, $\mathbf{S}_{\Omega}^{(a,b)} \in \mathcal{B}(L_p(\Omega; \mathbb{B}))$ . When $\Omega$ is of class $C^1$ and $1 , by Proposition 9(iv), <math>\mathbf{S}_{\Omega}^{(a,b)}$ is a Fredholm operator with index 0. In particular, it possesses a finite-dimensional kernel [28, Th. 2.22].

Definition 24. The Bergman kernel of the space $\mathcal{A}^2_{(a,b)}(\Omega;\mathbb{B})$ with coefficient $A \in \mathbb{B}$ is defined by <span id="page-13-1"></span> $$\mathscr{K}_{\Omega}^{(a,b)}(A;z,\zeta) := \operatorname{Sc}(A)K(z,\zeta) + \operatorname{Vec}(A)L(z,\zeta), \quad z,\zeta \in \Omega.$$ (38) The following reproducing property holds: <span id="page-13-2"></span> $$W(z) = \iint_{\Omega} \mathscr{K}_{\Omega}^{(a,b)}(W(\zeta); z, \zeta) dA_{\zeta}. \tag{39}$$ This definition for the bicomplex Bergman kernel of the Vekua equation was introduced in [32] for the case of the space of classical $L_2$ -solutions of the main Vekua equation (defined in Section 7). Because the space is separable, it admits a countable orthonormal basis $\{\Phi_n\}_{n=0}^{\infty}$ [14, Ch. I, Prop. 4.16]. Since $\mathscr{K}_{\Omega}^{(a,b)}(A;\cdot,\zeta) \in \mathcal{A}^2_{(a,b)}(\Omega;\mathbb{B})$ for all $A \in \mathbb{B}$ , $\zeta \in \Omega$ , it possesses a Fourier series in terms of $\{\Phi_n\}_{n=0}^{\infty}$ , and the form of its Fourier coefficients can be obtained in terms of the coefficient A.