Boundary value problems for analytic functions in the class of Cauchy-type integrals with density in

We study the Riemann boundary value problem , for analytic functions in the class of analytic functions represented by the Cauchy-type integrals with density in the spaces with variable exponent. We consider both the case when the coefficient is piecewise continuous and it may be of a more general nature, admitting its oscillation. The explicit formulas for solutions in the variable exponent setting are given. The related singular integral equations in the same setting are also investigated. As an application there is derived some extension of the Szegö-Helson theorem to the case of variable exponents.Peer Reviewe

Maximal, potential and singular operators in the local "complementary" variable exponent Morrey type spaces

We consider local "complementary" generalized Morrey spaces M-c({x0})p(.).omega (Omega) in which the p-means of function are controlled over Omega \ B(x(0), r) instead of B(x(0), r), where Omega subset of R-n is a bounded open set, p(x) is a variable exponent, and no monotonicity type condition is imposed onto the function omega(r) defining the "complementary" Morrey-type norm. In the case where omega is a power function, we reveal the relation of these spaces to weighted Lebesgue spaces. In the general case we prove the boundedness of the Hardy-Littlewood maximal operator and Calderon-Zygmund singular operators with standard kernel, in such spaces. We also prove a Sobolev type M-c({x0})p(.).omega (Omega) -> M-c({x0})p(.).omega (Omega)-theorem for the potential operators I-alpha(.), also of variable order. In all the cases the conditions for the boundedness are given it terms of Zygmund-type integral inequalities-on omega(r), which do not assume any assumption on monotonicity of omega(r).Science Development Foundation under the President of the Republic of Azerbaijan [EIF-2010-1(1)-40/06-1]; Scientific and Technological Research Council of Turkey (TUBITAK) [110T695]info:eu-repo/semantics/publishedVersio

Some sharp inequalities for integral operators with homogeneous kernel

One goal of this paper is to show that a big number of inequalities for functions in L-p(R+), p >= 1, proved from time to time in journal publications are particular cases of some known general results for integral operators with homogeneous kernels including, in particular, the statements on sharp constants. Some new results are also included, e.g. the similar general equivalence result is proved and applied for 0 < p < 1. Some useful new variants of these results are pointed out and a number of known and new Hardy-Hilbert type inequalities are derived. Moreover, a new Polya-Knopp (geometric mean) inequality is derived and applied. The constants in all inequalities in this paper are sharp

Maximal operator in variable exponent generalized morrey spaces on quasi-metric measure space

We consider generalized Morrey spaces on quasi-metric measure spaces , in general unbounded, with variable exponent p(x) and a general function defining the Morrey-type norm. No linear structure of the underlying space X is assumed. The admission of unbounded X generates problems known in variable exponent analysis. We prove the boundedness results for maximal operator known earlier only for the case of bounded sets X. The conditions for the boundedness are given in terms of the so called supremal inequalities imposed on the function , which are weaker than Zygmund-type integral inequalities often used for characterization of admissible functions . Our conditions do not suppose any assumption on monotonicity of in r

On a class of sublinear operators in variable exponent Morrey-type spaces

For a class of sublinear operators, we find conditions on the variable exponent Morrey-type space L-p(.),L-q,L-omega(.,L-.)(R-n) ensuring the boundedness in this space. A priori assumptions on this class are that the operators are bounded in L-p(.)(R-n) and satisfy some size condition. This class includes in particular the maximal operator, singular operators with the standard kernel, and the Hardy operators. Wealso prove embedding of variable exponent Morrey-type spaces into weighted L-p(.)-spaces.United Arab Emirates University, Al Ain, United Arab Emirates [G00002994]; Russian Foundation for Basic ResearchRussian Foundation for Basic Research (RFBR) [19-01-00223, 20-51-46003]; TUBITAKTurkiye Bilimsel ve Teknolojik Arastirma Kurumu (TUBITAK) [20-51-46003

Hardy inequality in variable exponent Lebesgue spaces

Diening L, Samko SG. Hardy inequality in variable exponent Lebesgue spaces. Fractional Calculus &amp; Applied Analysis. An International Journal for Theory and Applications. 2007;10(1):1-18

On potentials in generalized Hölder spaces over uniform domains in R

Abstract We show that Riesz-type potential operators of order α over uniform domains Ω in R n map the subspace H λ 0 (Ω) of functions in Hölder space H λ (Ω) vanishing on ∂Ω, into the space H λ+α (Ω), if λ + α ≤ 1. This is proved in a more general setting of generalized Hölder spaces with a given dominant of continuity modulus. Statements of such a kind are known for instance for the whole space R n or more generally for metric measure spaces with cancellation property. In the case of domains in R n when the cancellation property fails, our proofs are based on a special treatment of potential of a constant function

On potentials in generalized Hölder spaces over uniform domains in Rn\Bbb R^n

Diening L, Samko SG. On potentials in generalized Hölder spaces over uniform domains in $\Bbb R^n$. Revista Matemática Complutense. 2011;24(2):357-373

Integral equations of the first kind of Sonine type

A Volterra integral equation of the first kind Kφ(x):≡∫−∞xk(x−t)φ(t)dt=f(x) with a locally integrable kernel k(x)∈L1loc(ℝ+1) is called Sonine equation if there exists another locally integrable kernel ℓ(x) such that ∫0xk(x−t)ℓ(t)dt≡1 (locally integrable divisors of the unit, with respect to the operation of convolution). The formal inversion φ(x)=(d/dx)∫0xℓ(x−t)f(t)dt is well known, but it does not work, for example, on solutions in the spaces X=Lp(ℝ1) and is not defined on the whole range K(X). We develop many properties of Sonine kernels which allow us—in a very general case—to construct the real inverse operator, within the framework of the spaces Lp(ℝ1), in Marchaud form: K−1f(x)=ℓ(∞)f(x)+∫0∞ℓ′(t)[f(x−t)−f(x)]dt with the interpretation of the convergence of this hypersingular integral in Lp-norm. The description of the range K(X) is given; it already requires the language of Orlicz spaces even in the case when X is the Lebesgue space Lp(ℝ1)