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Lanzhe Liu - One of the best experts on this subject based on the ideXlab platform.

Alexei Yu Karlovich - One of the best experts on this subject based on the ideXlab platform.

  • on a weighted Singular Integral Operator with shifts and slowly oscillating data
    Complex Analysis and Operator Theory, 2016
    Co-Authors: Alexei Yu Karlovich, Yu I Karlovich, Amarino B Lebre
    Abstract:

    Let \(\alpha ,\beta \) be orientation-preserving diffeomorphism (shifts) of \(\mathbb {R}_+=(0,\infty )\) onto itself with the only fixed points \(0\) and \(\infty \) and \(U_\alpha ,U_\beta \) be the isometric shift Operators on \(L^p(\mathbb {R}_+)\) given by \(U_\alpha f=(\alpha ')^{1/p}(f\circ \alpha )\), \(U_\beta f=(\beta ')^{1/p}(f\circ \beta )\), and \(P_2^\pm =(I\pm S_2)/2\) where $$\begin{aligned} (S_2 f)(t):=\frac{1}{\pi i}\int \limits _0^\infty \left( \frac{t}{\tau }\right) ^{1/2-1/p}\frac{f(\tau )}{\tau -t}\,d\tau , \quad t\in \mathbb {R}_+, \end{aligned}$$ is the weighted Cauchy Singular Integral Operator. We prove that if \(\alpha ',\beta '\) and \(c,d\) are continuous on \(\mathbb {R}_+\) and slowly oscillating at \(0\) and \(\infty \), and $$\begin{aligned} \limsup _{t\rightarrow s}|c(t)|<1, \quad \limsup _{t\rightarrow s}|d(t)|<1, \quad s\in \{0,\infty \}, \end{aligned}$$ then the Operator \((I-cU_\alpha )P_2^++(I-dU_\beta )P_2^-\) is Fredholm on \(L^p(\mathbb {R}_+)\) and its index is equal to zero. Moreover, its regularizers are described.

  • On a Weighted Singular Integral Operator with Shifts and Slowly Oscillating Data
    Complex Analysis and Operator Theory, 2016
    Co-Authors: Alexei Yu Karlovich, Yu I Karlovich, Amarino B Lebre
    Abstract:

    Let $$\alpha ,\beta $$ α , β be orientation-preserving diffeomorphism (shifts) of $$\mathbb {R}_+=(0,\infty )$$ R + = ( 0 , ∞ ) onto itself with the only fixed points $$0$$ 0 and $$\infty $$ ∞ and $$U_\alpha ,U_\beta $$ U α , U β be the isometric shift Operators on $$L^p(\mathbb {R}_+)$$ L p ( R + ) given by $$U_\alpha f=(\alpha ')^{1/p}(f\circ \alpha )$$ U α f = ( α ′ ) 1 / p ( f ∘ α ) , $$U_\beta f=(\beta ')^{1/p}(f\circ \beta )$$ U β f = ( β ′ ) 1 / p ( f ∘ β ) , and $$P_2^\pm =(I\pm S_2)/2$$ P 2 ± = ( I ± S 2 ) / 2 where $$\begin{aligned} (S_2 f)(t):=\frac{1}{\pi i}\int \limits _0^\infty \left( \frac{t}{\tau }\right) ^{1/2-1/p}\frac{f(\tau )}{\tau -t}\,d\tau , \quad t\in \mathbb {R}_+, \end{aligned}$$ ( S 2 f ) ( t ) : = 1 π i ∫ 0 ∞ t τ 1 / 2 - 1 / p f ( τ ) τ - t d τ , t ∈ R + , is the weighted Cauchy Singular Integral Operator. We prove that if $$\alpha ',\beta '$$ α ′ , β ′ and $$c,d$$ c , d are continuous on $$\mathbb {R}_+$$ R + and slowly oscillating at $$0$$ 0 and $$\infty $$ ∞ , and $$\begin{aligned} \limsup _{t\rightarrow s}|c(t)|

  • on a weighted Singular Integral Operator with shifts and slowly oscillating data
    arXiv: Functional Analysis, 2015
    Co-Authors: Alexei Yu Karlovich, Yu I Karlovich, Amarino B Lebre
    Abstract:

    Let $\alpha,\beta$ be orientation-preserving diffeomorphism (shifts) of $\mathbb{R}_+=(0,\infty)$ onto itself with the only fixed points $0$ and $\infty$ and $U_\alpha,U_\beta$ be the isometric shift Operators on $L^p(\mathbb{R}_+)$ given by $U_\alpha f=(\alpha')^{1/p}(f\circ\alpha)$, $U_\beta f=(\beta')^{1/p}(f\circ\beta)$, and $P_2^\pm=(I\pm S_2)/2$ where \[ (S_2 f)(t):=\frac{1}{\pi i}\int\limits_0^\infty \left(\frac{t}{\tau}\right)^{1/2-1/p}\frac{f(\tau)}{\tau-t}\,d\tau, \quad t\in\mathbb{R}_+, \] is the weighted Cauchy Singular Integral Operator. We prove that if $\alpha',\beta'$ and $c,d$ are continuous on $\mathbb{R}_+$ and slowly oscillating at $0$ and $\infty$, and \[ \limsup_{t\to s}|c(t)|<1, \quad \limsup_{t\to s}|d(t)|<1, \quad s\in\{0,\infty\}, \] then the Operator $(I-cU_\alpha)P_2^++(I-dU_\beta)P_2^-$ is Fredholm on $L^p(\mathbb{R}_+)$ and its index is equal to zero. Moreover, its regularizers are described.

  • the cauchy Singular Integral Operator on weighted variable lebesgue spaces
    Concrete Operators Spectral Theory Operators in Harmonic Analysis and Approximation. 22nd International Workshop in Operator Theory and its Applicatio, 2014
    Co-Authors: Alexei Yu Karlovich, Ilya M Spitkovsky
    Abstract:

    Let p: ℝ → (1,∞) be a globally log-Holder continuous variable exponent and w: ℝ →[0,∞] be a weight. We prove that the Cauchy Singular Integral Operator s is bounded on the weighted variable Lebesgue space L p(.)(ℝ,w)= {f:f wϵL p(.)(ℝ)} if and only if the weight w satisfies

  • Sufficient Conditions for Fredholmness of Singular Integral Operators with Shifts and Slowly Oscillating Data
    Integral Equations and Operator Theory, 2011
    Co-Authors: Alexei Yu Karlovich, Yu I Karlovich, Amarino B Lebre
    Abstract:

    Suppose α is an orientation preserving diffeomorphism (shift) of $${{\mathbb{R}}_+=(0,\infty)}$$ onto itself with the only fixed points 0 and ∞. We establish sufficient conditions for the Fredholmness of the Singular Integral Operator with shift $$(aI-bW_\alpha)P_++(cI-dW_\alpha)P_-$$ acting on $${L^p({\mathbb{R}}_+)}$$ with 1 

Dazhao Chen - One of the best experts on this subject based on the ideXlab platform.

Hussain Al-qassem - One of the best experts on this subject based on the ideXlab platform.

Amarino B Lebre - One of the best experts on this subject based on the ideXlab platform.

  • On a Weighted Singular Integral Operator with Shifts and Slowly Oscillating Data
    Complex Analysis and Operator Theory, 2016
    Co-Authors: Alexei Yu Karlovich, Yu I Karlovich, Amarino B Lebre
    Abstract:

    Let $$\alpha ,\beta $$ α , β be orientation-preserving diffeomorphism (shifts) of $$\mathbb {R}_+=(0,\infty )$$ R + = ( 0 , ∞ ) onto itself with the only fixed points $$0$$ 0 and $$\infty $$ ∞ and $$U_\alpha ,U_\beta $$ U α , U β be the isometric shift Operators on $$L^p(\mathbb {R}_+)$$ L p ( R + ) given by $$U_\alpha f=(\alpha ')^{1/p}(f\circ \alpha )$$ U α f = ( α ′ ) 1 / p ( f ∘ α ) , $$U_\beta f=(\beta ')^{1/p}(f\circ \beta )$$ U β f = ( β ′ ) 1 / p ( f ∘ β ) , and $$P_2^\pm =(I\pm S_2)/2$$ P 2 ± = ( I ± S 2 ) / 2 where $$\begin{aligned} (S_2 f)(t):=\frac{1}{\pi i}\int \limits _0^\infty \left( \frac{t}{\tau }\right) ^{1/2-1/p}\frac{f(\tau )}{\tau -t}\,d\tau , \quad t\in \mathbb {R}_+, \end{aligned}$$ ( S 2 f ) ( t ) : = 1 π i ∫ 0 ∞ t τ 1 / 2 - 1 / p f ( τ ) τ - t d τ , t ∈ R + , is the weighted Cauchy Singular Integral Operator. We prove that if $$\alpha ',\beta '$$ α ′ , β ′ and $$c,d$$ c , d are continuous on $$\mathbb {R}_+$$ R + and slowly oscillating at $$0$$ 0 and $$\infty $$ ∞ , and $$\begin{aligned} \limsup _{t\rightarrow s}|c(t)|

  • on a weighted Singular Integral Operator with shifts and slowly oscillating data
    Complex Analysis and Operator Theory, 2016
    Co-Authors: Alexei Yu Karlovich, Yu I Karlovich, Amarino B Lebre
    Abstract:

    Let \(\alpha ,\beta \) be orientation-preserving diffeomorphism (shifts) of \(\mathbb {R}_+=(0,\infty )\) onto itself with the only fixed points \(0\) and \(\infty \) and \(U_\alpha ,U_\beta \) be the isometric shift Operators on \(L^p(\mathbb {R}_+)\) given by \(U_\alpha f=(\alpha ')^{1/p}(f\circ \alpha )\), \(U_\beta f=(\beta ')^{1/p}(f\circ \beta )\), and \(P_2^\pm =(I\pm S_2)/2\) where $$\begin{aligned} (S_2 f)(t):=\frac{1}{\pi i}\int \limits _0^\infty \left( \frac{t}{\tau }\right) ^{1/2-1/p}\frac{f(\tau )}{\tau -t}\,d\tau , \quad t\in \mathbb {R}_+, \end{aligned}$$ is the weighted Cauchy Singular Integral Operator. We prove that if \(\alpha ',\beta '\) and \(c,d\) are continuous on \(\mathbb {R}_+\) and slowly oscillating at \(0\) and \(\infty \), and $$\begin{aligned} \limsup _{t\rightarrow s}|c(t)|<1, \quad \limsup _{t\rightarrow s}|d(t)|<1, \quad s\in \{0,\infty \}, \end{aligned}$$ then the Operator \((I-cU_\alpha )P_2^++(I-dU_\beta )P_2^-\) is Fredholm on \(L^p(\mathbb {R}_+)\) and its index is equal to zero. Moreover, its regularizers are described.

  • on a weighted Singular Integral Operator with shifts and slowly oscillating data
    arXiv: Functional Analysis, 2015
    Co-Authors: Alexei Yu Karlovich, Yu I Karlovich, Amarino B Lebre
    Abstract:

    Let $\alpha,\beta$ be orientation-preserving diffeomorphism (shifts) of $\mathbb{R}_+=(0,\infty)$ onto itself with the only fixed points $0$ and $\infty$ and $U_\alpha,U_\beta$ be the isometric shift Operators on $L^p(\mathbb{R}_+)$ given by $U_\alpha f=(\alpha')^{1/p}(f\circ\alpha)$, $U_\beta f=(\beta')^{1/p}(f\circ\beta)$, and $P_2^\pm=(I\pm S_2)/2$ where \[ (S_2 f)(t):=\frac{1}{\pi i}\int\limits_0^\infty \left(\frac{t}{\tau}\right)^{1/2-1/p}\frac{f(\tau)}{\tau-t}\,d\tau, \quad t\in\mathbb{R}_+, \] is the weighted Cauchy Singular Integral Operator. We prove that if $\alpha',\beta'$ and $c,d$ are continuous on $\mathbb{R}_+$ and slowly oscillating at $0$ and $\infty$, and \[ \limsup_{t\to s}|c(t)|<1, \quad \limsup_{t\to s}|d(t)|<1, \quad s\in\{0,\infty\}, \] then the Operator $(I-cU_\alpha)P_2^++(I-dU_\beta)P_2^-$ is Fredholm on $L^p(\mathbb{R}_+)$ and its index is equal to zero. Moreover, its regularizers are described.

  • Sufficient Conditions for Fredholmness of Singular Integral Operators with Shifts and Slowly Oscillating Data
    Integral Equations and Operator Theory, 2011
    Co-Authors: Alexei Yu Karlovich, Yu I Karlovich, Amarino B Lebre
    Abstract:

    Suppose α is an orientation preserving diffeomorphism (shift) of $${{\mathbb{R}}_+=(0,\infty)}$$ onto itself with the only fixed points 0 and ∞. We establish sufficient conditions for the Fredholmness of the Singular Integral Operator with shift $$(aI-bW_\alpha)P_++(cI-dW_\alpha)P_-$$ acting on $${L^p({\mathbb{R}}_+)}$$ with 1