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Alexei Yu. Karlovich

Publications and source records attributed to Alexei Yu. Karlovich.

36 records · Page 2Linked to original sources
arXiv2010

Necessary Conditions for Fredholmness of Singular Integral Operators with Shifts and Slowly Oscillating Data

Suppose $α$ is an orientation-preserving diffeomorphism (shift) of $\mR_+=(0,\infty)$ onto itself with the only fixed points $0$ and $\infty$. In \cite{KKLsufficiency} we found sufficient conditions for the Fredholmness of the singular integral operator with shift \[ (aI-bW_α)P_++(cI-dW_α)P_- \] acting on $L^p(\mR_+)$ with $1<p<\infty$, where $P_\pm=(I\pm S)/2$, $S$ is the Cauchy singular integral operator, and $W_αf=f\circα$ is the shift operator, under the assumptions that the coefficients $a,b,c,d$ and the derivative $α'$ of the shift are bounded and continuous on $\mR_+$ and may admit discontinuities of slowly oscillating type at $0$ and $\infty$. Now we prove that those conditions are also necessary.

math.FA↗
arXiv2010

Sufficient Conditions for Fredholmness of Singular Integral Operators with Shifts and Slowly Oscillating Data

Suppose $α$ is an orientation preserving diffeomorphism (shift) of $\mR_+=(0,\infty)$ onto itself with the only fixed points $0$ and $\infty$. We establish sufficient conditions for the Fredholmness of the singular integral operator \[ (aI-bW_α)P_++(cI-dW_α)P_- \] acting on $L^p(\mR_+)$ with $1<p<\infty$, where $P_\pm=(I\pm S)/2$, $S$ is the Cauchy singular integral operator, and $W_αf=f\circα$ is the shift operator, under the assumptions that the coefficients $a,b,c,d$ and the derivative $α'$ of the shift are bounded and continuous on $\mR_+$ and may admit discontinuities of slowly oscillating type at $0$ and $\infty$.

math.FA↗
arXiv2010

Singular integral operators on Nakano spaces with weights having finite sets of discontinuities

In 1968, Gohberg and Krupnik found a Fredholm criterion for singular integral operators of the form $aP+bQ$, where $a,b$ are piecewise continuous functions and $P,Q$ are complementary projections associated to the Cauchy singular integral operator, acting on Lebesgue spaces over Lyapunov curves. We extend this result to the case of Nakano spaces (also known as variable Lebesgue spaces) with certain weights having finite sets of discontinuities on arbitrary Carleson curves.

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arXiv2009

Finite Section Method for a Banach Algebra of Convolution Type Operators on $L^p(\mathbb{R})$ with Symbols Generated by $PC$ and $SO$

We prove the applicability of the finite section method to an arbitrary operator in the Banach algebra generated by the operators of multiplication by piecewise continuous functions and the convolution operators with symbols in the algebra generated by piecewise continuous and slowly oscillating Fourier multipliers on $L^p(\mathbb{R})$, $1<p<\infty$.

math.FA↗
arXiv2009

Singular integral operators on variable Lebesgue spaces with radial oscillating weights

We prove a Fredholm criterion for operators in the Banach algebra of singular integral operators with matrix piecewise continuous coefficients acting on a variable Lebesgue space with a radial oscillating weight over a logarithmic Carleson curve. The local spectra of these operators are massive and have a shape of spiralic horns depending on the value of the variable exponent, the spirality indices of the curve, and the Matuszewska-Orlicz indices of the weight at each point. These results extend (partially) the results of A. Böttcher, Yu. Karlovich, and V. Rabinovich for standard Lebesgue spaces to the case of variable Lebesgue spaces.

math.FA↗
arXiv2008

Singular Integral Operators on Variable Lebesgue Spaces over Arbitrary Carleson Curves

In 1968, Israel Gohberg and Naum Krupnik discovered that local spectra of singular integral operators with piecewise continuous coefficients on Lebesgue spaces $L^p(Γ)$ over Lyapunov curves have the shape of circular arcs. About 25 years later, Albrecht Böttcher and Yuri Karlovich realized that these circular arcs metamorphose to so-called logarithmic leaves with a median separating point when Lyapunov curves metamorphose to arbitrary Carleson curves. We show that this result remains valid in a more general setting of variable Lebesgue spaces $L^{p(\cdot)}(Γ)$ where $p:Γ\to(1,\infty)$ satisfies the Dini-Lipschitz condition. One of the main ingredients of the proof is a new sufficient condition for the boundedness of the Cauchy singular integral operator on variable Lebesgue spaces with weights related to oscillations of Carleson curves.

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arXiv2008

Maximal operators on variable Lebesgue spaces with weights related to oscillations of Carleson curves

We prove sufficient conditions for the boundedness of the maximal operator on variable Lebesgue spaces with weights $φ_{t,γ}(τ)=|(τ-t)^γ|$, where $γ$ is a complex number, over arbitrary Carleson curves. If the curve has different spirality indices at the point $t$ and $γ$ is not real, then $φ_{t,γ}$ is an oscillating weight lying beyond the class of radial oscillating weights considered recently by V. Kokilashvili, N. Samko, and S. Samko.

math.CA↗
arXiv2008

Asymptotics of Toeplitz Matrices with Symbols in Some Generalized Krein Algebras

Let $α,β\in(0,1)$ and \[ K^{α,β}:=\left\{a\in L^\infty(\T): \sum_{k=1}^\infty |\hat{a}(-k)|^2 k^{2α}<\infty, \sum_{k=1}^\infty |\hat{a}(k)|^2 k^{2β}<\infty \right\}. \] Mark Krein proved in 1966 that $K^{1/2,1/2}$ forms a Banach algebra. He also observed that this algebra is important in the asymptotic theory of finite Toeplitz matrices. Ten years later, Harold Widom extended earlier results of Gabor Szegő for scalar symbols and established the asymptotic trace formula \[ \operatorname{trace}f(T_n(a))=(n+1)G_f(a)+E_f(a)+o(1) \quad\text{as}\ n\to\infty \] for finite Toeplitz matrices $T_n(a)$ with matrix symbols $a\in K^{1/2,1/2}_{N\times N}$. We show that if $α+β\ge 1$ and $a\in K^{α,β}_{N\times N}$, then the Szegő-Widom asymptotic trace formula holds with $o(1)$ replaced by $o(n^{1-α-β})$.

math.FA↗
arXiv2007

Higher Order Asymptotic Formulas for Traces of Toeplitz Matrices with Symbols in Hölder-Zygmund Spaces

We prove a higher order asymptotic formula for traces of finite block Toeplitz matrices with symbols belonging to Hölder-Zygmund spaces. The remainder in this formula goes to zero very rapidly for very smooth symbols. This formula refine previous asymptotic trace formulas by Szegö and Widom and complement higher order asymptotic formulas for determinants of finite block Toeplitz matrices due to Böttcher and Silbermann.

math.FA↗
arXiv2007

On the interpolation constant for subadditive operators in Orlicz spaces

Let $1\le p<q\le\infty$ and let $T$ be a subadditive operator acting on $L^p$ and $L^q$. We prove that $T$ is bounded on the Orlicz space $L^ϕ$, where $ϕ^{-1}(u)=u^{1/p}ρ(u^{1/q-1/p})$ for some concave function $ρ$ and \[ \|T\|_{L^ϕ\to L^ϕ}\le C\max\{\|T\|_{L^p\to L^p},\|T\|_{L^q\to L^q}\}. \] The interpolation constant $C$, in general, is less than 4 and, in many cases, we can give much better estimates for $C$. In particular, if $p=1$ and $q=\infty$, then the classical Orlicz interpolation theorem holds for subadditive operators with the interpolation constant C=1. These results generalize our results for linear operators obtained in \cite{KM01}.

math.FA↗
arXiv2007

Semi-Fredholm singular integral operators with piecewise continuous coefficients on weighted variable Lebesgue spaces are Fredholm

Suppose $Γ$ is a Carleson Jordan curve with logarithmic whirl points, $\varrho$ is a Khvedelidze weight, $p:Γ\to(1,\infty)$ is a continuous function satisfying $|p(τ)-p(t)|\le -\mathrm{const}/\log|τ-t|$ for $|τ-t|\le 1/2$, and $L^{p(\cdot)}(Γ,\varrho)$ is a weighted generalized Lebesgue space with variable exponent. We prove that all semi-Fredholm operators in the algebra of singular integral operators with $N\times N$ matrix piecewise continuous coefficients are Fredholm on $L_N^{p(\cdot)}(Γ,\varrho)$.

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arXiv2003

Fredholmness of singular integral operators with piecewise continuous coefficients on weighted Banach function spaces

We prove necessary conditions for Fredholmness of singular integral operators with piecewise continuous coefficients on weighted Banach function spaces. These conditions are formulated in terms of indices of submultiplicative functions associated with local properties of the space, of the curve, and of the weight. As an example, we consider weighted Nakano spaces $L^{p(\cdot)}_w$ (weighted Lebesgue spaces with variable exponent). Moreover, our necessary conditions become also sufficient for weighted Nakano spaces over nice curves whenever $w$ is a Khvedelidze weight, and the variable exponent $p(t)$ satisfies the estimate $|p(τ)-p(t)|\le A/(-\log|τ-t|)$.

math.FA↗
arXiv2003

Algebras of functions with Fourier coefficients in weighted Orlicz sequence spaces

We prove that the set of all integrable functions whose sequences of negative (resp. nonnegative) Fourier coefficients belong to $\ell^1\cap\ell^Φ_{ϕ,w}$ (resp. to $\ell^1\cap\ell^Ψ_{ψ,\varrho}$), where $\ell^Φ_{ϕ,w}$ and $\ell^Ψ_{ψ,\varrho}$ are two-weighted Orlicz sequence spaces, forms an algebra under pointwise multiplication whenever the weight sequences \[ ϕ=\{ϕ_n\},\quad ψ=\{ψ_n\},\quad w=\{w_n\},\quad \varrho=\{\varrho_n\} \] increase and satisfy the $Δ_2$-condition.

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