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Let X be a Banach function space on the unit circle T , let X ′ be its associate space, and let H[X] and H [ X ′ ] be the abstract Hardy spaces built upon X and X ′ , respectively. Suppose that the Riesz projection P is bounded on X and a ∈ L ∞ \ { 0 } . We show that P is bounded on X ′ . So, we can consider the Toeplitz operators T (a) f = P (a f) and T (a ¯) g = P (a ¯ g) on H[X] and H [ X ′ ] , respectively. In our previous paper, we have shown that if X is not separable, then one cannot rephrase Coburn's lemma as in the case of classical Hardy spaces H p , 1 < p < ∞ , and guarantee that T(a) has a trivial kernel or a dense range on H[X]. The first main result of the present paper is the following extension of Coburn's lemma: the kernel of T(a) or the kernel of T (a ¯) is trivial. The second main result is a generalisation of the Hartman–Wintner–Simonenko theorem saying that if T(a) is normally solvable on the space H[X], then 1 / a ∈ L ∞ . [ABSTRACT FROM AUTHOR]

We study the weighted boundedness of the multi-dimensional Hardy-type and singular operators in the generalized Morrey spaces [ABSTRACT FROM AUTHOR]

To each Dirac type operator there is assigned a formally exact elliptic complex of length 2. We study the Neumann problem for this complex in a bounded domain with smooth boundary in R, as it was formulated by Spencer in 1957. [ABSTRACT FROM AUTHOR]

The Fredholm property of some linear infinite dimensional difference operators is studied. In the case corresponding to discretization of differential equations on the real axis, the index of the corresponding operators is computed and solvability conditions for the nonhomogeneous problem are established. In the multi-dimensional case, conditions of the normal solvability of the corresponding discrete operators are formulated in terms of limiting problems. The results on the location of the spectrum and the solvability conditions allow various applications to linear and nonlinear problems. [ABSTRACT FROM AUTHOR]

We discuss explicit boundary value problems for solutions of the Fueter equation in $${\mathbb{R}}^{4}$$ which are normally solvable. The results extend to nonlinear first order elliptic systems. [ABSTRACT FROM AUTHOR]