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

Kokilashvili, Vakhtang; Paatashvili, V.; Samko, Stefan

doi:10.1155/bvp.2005.43

Cited by 29 publications

(32 citation statements)

References 20 publications

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“…Thus D is a Smirnov domain [13,Corollary 3.4]. This gives us the right to apply the Smirnov theorem (see Section 2).…”

Section: Conditions Of the Coincidence Of The Classesmentioning

confidence: 94%

“…In the last decade, an intensive development of the theory of Lebesgue spaces with a variable exponent has made it possible to investigate boundary value problems of analytic functions and mathematical physics formulated in more advantageous terms, taking into account the local behavior of the given and unknown functions (see, e.g., [2,6,7,[12][13][14][15][16][17][18]). In particular, in [13], a problem of linear conjugation is investigated in the class of functions representable by a Cauchy type integrals with density from the variable Lebesgue spaces, while in [12] the Dirichlet problem is studied in the class of harmonic functions which are the real parts of functions from the above-mentioned classes.…”

Section: Introductionmentioning

confidence: 99%

“…In particular, in [13], a problem of linear conjugation is investigated in the class of functions representable by a Cauchy type integrals with density from the variable Lebesgue spaces, while in [12] the Dirichlet problem is studied in the class of harmonic functions which are the real parts of functions from the above-mentioned classes. As to the boundary, it is assumed that Γ is a piecewise-Lyapunov curve.…”

Section: Introductionmentioning

confidence: 99%

“…The set of all such functions φ is called by the weighted class of Cauchy type integrals with density from L p( · ) (Γ) and is denoted by K p( · ) (D; ω) as different from the set of Cauchy type integrals with density from L p( · ) (Γ; ω) denoted by K p( · ) (Γ; ω) [13]. When a weight function ω ∈ W p( · ) (Γ), i.e., the Cauchy singular operator is continuous in L p( · ) (Γ; ω), we show that K p( · ) (D; ω) and K p( · ) (Γ; ω) coincide for the wide class of curves Γ and functions p(t) (see Theorem 1 and its corollary below).…”

Section: Introductionmentioning

confidence: 99%

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The Riemann–Hilbert Problem in Weighted Classes of Cauchy Type Integrals with Density from L P( · )(Γ)

Kokilashvili¹,

Paatashvili²

2008

Complex anal.oper. theory

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We consider the Riemann-Hilbert problem in the following setting: find a function φ ∈ K p( · ) (D; ω) whose boundary values φ + (t) satisfy the condition Re[(a(t) + ib(t))φ + (t)] = c(t) a.e. on Γ. Here D is a simply connected domain bounded by a simple closed curve Γ, andis a weight function and (KΓϕ)(z) is a Cauchy type integral whose density ϕ is integrable with a variable exponent p(t). It is assumed that Γ is a piecewiseLyapunov curve without zero angles, ω(z) is an arbitrary power function and p(t) satisfies the Log-Hölder condition. The solvability conditions are established and solutions are constructed. These solutions largely depend on the coefficients a, b, c, the weight ω, on the values of p(t) at the angular points of Γ and on the values of angles at these points. Mathematics Subject Classification (2000). Primary 47B38, 30E20, 30E25; Secondary 42B20, 45P05.

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“…Thus D is a Smirnov domain [13,Corollary 3.4]. This gives us the right to apply the Smirnov theorem (see Section 2).…”

Section: Conditions Of the Coincidence Of The Classesmentioning

confidence: 94%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The Riemann–Hilbert Problem in Weighted Classes of Cauchy Type Integrals with Density from L P( · )(Γ)

Kokilashvili¹,

Paatashvili²

2008

Complex anal.oper. theory

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“…Similar to the case of the constant p, the Fredholm theory of the mentioned operators in spaces related to L p(·) has also a big interest. With respect to one-dimensional singular integral operators in variable exponent Lebesgue spaces we refer, for instance, to [17][18][19][20][21][22][23][24][25][26]41].…”

Section: Introductionmentioning

confidence: 99%

Pseudodifferential Operators Approach to Singular Integral Operators in Weighted Variable Exponent Lebesgue Spaces on Carleson Curves

Rabinovich

2010

Integr. Equ. Oper. Theory

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The main results of the paper are: (1) The boundedness of singular integral operators in the variable exponent Lebesgue spaces L p(·) (Γ, w) on a class of composed Carleson curves Γ where the weights w have a finite set of oscillating singularities. The proof of this result is based on the boundedness of Mellin pseudodifferential operators on the spaces L p(·) (R+, dμ) where dμ is an invariant measure on multiplicative group R+ = {r ∈ R : r > 0}.(2) Criterion of local invertibility of singular integral operators with piecewise slowly oscillating coefficients acting on L p(·) (Γ, w) spaces. We obtain this criterion from the corresponding criteria of local invertibility at the point 0 of Mellin pseudodifferential operators on R+ and local invertibility of singular integral operators on R.(3) Criterion of Fredholmness of singular integral operators in the variable exponent Lebesgue spaces L p(·) (Γ, w) where Γ belongs to a class of composed Carleson curves slowly oscillating at the nodes, and the weight w has a finite set of slowly oscillating singularities. Mathematics Subject Classification (2000). Primary 47G30.

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On the interpolation constants for variable Lebesgue spaces

Karlovych

Shargorodsky

2023

Mathematische Nachrichten

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For θ∈false(0,1false)$\theta \in (0,1)$ and variable exponents p0(·),q0(·)$p_0(\cdot ),q_0(\cdot )$ and p1(·),q1(·)$p_1(\cdot ),q_1(\cdot )$ with values in [1, ∞], let the variable exponents pθ(·),qθ(·)$p_\theta (\cdot ),q_\theta (\cdot )$ be defined by 1/pθ(·):=(1−θ)/p0(·)goodbreak+θ/p1(·),1em1/qθ(·):=(1−θ)/q0(·)goodbreak+θ/q1(·).$$\begin{equation*} 1/p_\theta (\cdot ):=(1-\theta )/p_0(\cdot )+\theta /p_1(\cdot ), \quad 1/q_\theta (\cdot ):=(1-\theta )/q_0(\cdot )+\theta /q_1(\cdot ). \end{equation*}$$The Riesz–Thorin–type interpolation theorem for variable Lebesgue spaces says that if a linear operator T acts boundedly from the variable Lebesgue space Lpjfalse(·false)$L^{p_j(\cdot )}$ to the variable Lebesgue space Lqjfalse(·false)$L^{q_j(\cdot )}$ for j=0,1$j=0,1$, then false∥Tfalse∥Lpθ(·)→Lqθ(·)badbreak≤Cfalse∥Tfalse∥Lp0(·)→Lq0(·)1−θfalse∥Tfalse∥Lp1(·)→Lq1(·)θ,$$\begin{equation*} \Vert T\Vert _{L^{p_\theta (\cdot )}\rightarrow L^{q_\theta (\cdot )}} \le C \Vert T\Vert _{L^{p_0(\cdot )}\rightarrow L^{q_0(\cdot )}}^{1-\theta } \Vert T\Vert _{L^{p_1(\cdot )}\rightarrow L^{q_1(\cdot )}}^{\theta }, \end{equation*}$$where C is an interpolation constant independent of T. We consider two different modulars ϱmax(·)$\varrho ^{\max }(\cdot )$ and ϱsum(·)$\varrho ^{\rm sum}(\cdot )$ generating variable Lebesgue spaces and give upper estimates for the corresponding interpolation constants Cmax and Csum, which imply that Cmax≤2$C_{\rm max}\le 2$ and Csum≤4$C_{\rm sum}\le 4$, as well as, lead to sufficient conditions for Cmax=1$C_{\rm max}=1$ and Csum=1$C_{\rm sum}=1$. We also construct an example showing that, in many cases, our upper estimates are sharp and the interpolation constant is greater than one, even if one requires that pj(·)=qj(·)$p_j(\cdot )=q_j(\cdot )$, j=0,1$j=0,1$ are Lipschitz continuous and bounded away from one and infinity (in this case, ϱmax(·)=ϱsum(·)$\varrho ^{\rm max}(\cdot )=\varrho ^{\rm sum}(\cdot )$).

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

Cited by 29 publications

References 20 publications

The Riemann–Hilbert Problem in Weighted Classes of Cauchy Type Integrals with Density from L P( · )(Γ)

The Riemann–Hilbert Problem in Weighted Classes of Cauchy Type Integrals with Density from L P( · )(Γ)

Pseudodifferential Operators Approach to Singular Integral Operators in Weighted Variable Exponent Lebesgue Spaces on Carleson Curves

On the interpolation constants for variable Lebesgue spaces

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