arXiv subjects

Amarino B. Lebre

Publications and source records attributed to Amarino B. Lebre.

5 recordsLinked to original sources

arXiv2017

Semi-Fredholmness of weighted singular integral operators with shifts and slowly oscillating data

Let $α,β$ be orientation-preserving homeomorphisms of $[0,\infty]$ onto itself, which have only two fixed points at $0$ and $\infty$, and whose restrictions to $\mathbb{R}_+=(0,\infty)$ are diffeomorphisms, and let $U_α,U_β$ be the corresponding isometric shift operators on the space $L^p(\mathbb{R}_+)$ given by $U_μf=(μ')^{1/p}(f\circμ)$ for $μ\in\{α,β\}$. We prove sufficient conditions for the right and left Fredholmness on $L^p(\mathbb{R}_+)$ of singular integral operators of the form $A_+P_γ^++A_-P_γ^-$, where $P_γ^\pm=(I\pm S_γ)/2$, $S_γ$ is a weighted Cauchy singular integral operator, $A_+=\sum_{k\in\mathbb{Z}}a_kU_α^k$ and $A_-=\sum_{k\in\mathbb{Z}}b_kU_β^k$ are operators in the Wiener algebras of functional operators with shifts. We assume that the coefficients $a_k,b_k$ for $k\in\mathbb{Z}$ and the derivatives of the shifts $α',β'$ are bounded continuous functions on $\mathbb{R}_+$ which may have slowly oscillating discontinuities at $0$ and $\infty$.

math.FA↗

arXiv2015

On a Weighted Singular Integral Operator with Shifts and Slowly Oscillating Data

Let $α,β$ be orientation-preserving diffeomorphism (shifts) of $\mathbb{R}_+=(0,\infty)$ onto itself with the only fixed points $0$ and $\infty$ and $U_α,U_β$ be the isometric shift operators on $L^p(\mathbb{R}_+)$ given by $U_αf=(α')^{1/p}(f\circα)$, $U_βf=(β')^{1/p}(f\circβ)$, and $P_2^\pm=(I\pm S_2)/2$ where \[ (S_2 f)(t):=\frac{1}{πi}\int\limits_0^\infty \left(\frac{t}τ\right)^{1/2-1/p}\frac{f(τ)}{τ-t}\,dτ, \quad t\in\mathbb{R}_+, \] is the weighted Cauchy singular integral operator. We prove that if $α',β'$ 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 $(I-cU_α)P_2^++(I-dU_β)P_2^-$ is Fredholm on $L^p(\mathbb{R}_+)$ and its index is equal to zero. Moreover, its regularizers are described.

math.FA↗

arXiv2014

On regularization of Mellin PDO's with slowly oscillating symbols of limited smoothness

We study Mellin pseudodifferential operators (shortly, Mellin PDO's) with symbols in the algebra $\widetilde{\mathcal{E}}(\mathbb{R}_+,V(\mathbb{R}))$ of slowly oscillating functions of limited smoothness introduced in \cite{K09}. We show that if $\mathfrak{a}\in\widetilde{\mathcal{E}}(\mathbb{R}_+,V(\mathbb{R}))$ does not degenerate on the "boundary" of $\mathbb{R}_+\times\mathbb{R}$ in a certain sense, then the Mellin PDO ${\rm Op}(\mathfrak{a})$ is Fredholm on the space $L^p$ for $p\in(1,\infty)$ and each its regularizer is of the form ${\rm Op}(\mathfrak{b})+K$ where $K$ is a compact operator on $L^p$ and $\mathfrak{b}$ is a certain explicitly constructed function in the same algebra $\widetilde{\mathcal{E}}(\mathbb{R}_+,V(\mathbb{R}))$ such that $\mathfrak{b}=1/\mathfrak{a}$ on the "boundary" of $\mathbb{R}_+\times\mathbb{R}$. This result complements a known Fredholm criterion from \cite{K09} for Mellin PDO's with symbols in the closure of $\widetilde{\mathcal{E}}(\mathbb{R}_+,V(\mathbb{R}))$.

math.FA↗

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↗