q(x) \qquad \mbox{ in } \partial \Omega \cap \{x\in\overline{\Omega}\colon n-sp(x,x) >0\}, \]then the inequality $$ \Vert f\Vert _{\scriptstyle L^{q(\cdot)}(\partial \Omega )} \leq C \left\{ \Vert f\Vert _{\scriptstyle L^{\bar{p}(\cdot)}(\Omega )}+ [f]_{s,p(\cdot,\cdot)} \right\} $$ holds. Here $\bar{p}(x)=p(x,x)$ and $\lbrack f\rbrack_{s,p(\cdot,\cdot)} $ denotes the fractional seminorm with variable exponent, that is given by \begin{equation*} \lbrack f\rbrack_{s,p(\cdot,\cdot)} := \inf \left\{ \lambda >0\colon \int_{\Omega}\int_{\Omega }\frac{|f(x)-f(y)|^{p(x,y)}}{\lambda ^{p(x,y)} |x-y|^{n+sp(x,y)}}dxdy<1\right\} \end{equation*}and $\Vert f\Vert _{\scriptstyle L^{q(\cdot)}(\partial \Omega )}$ and $\Vert f\Vert _{\scriptstyle L^{\bar{p}(\cdot)}(\Omega )}$ are the usual Lebesgue norms with variable exponent.]]>
0$, we examine the relationship between the $\varepsilon$-pseudospectrum $\Lambda_{\varepsilon}(A,a)$ of $a\in A$, and $\varepsilon$-pseudospectra of $pap\in pAp$ and $qaq\in qAq$. We also extend this study by considering a finite number of idempotents $p_{1},\cdots,p_{n}$, as well as an arbitrary family of idempotents satisfying certain conditions.]]>
-1)$ are deeply intertwined with the Laplacian on compact rank one symmetric spaces. They represent the {\it spherical} or zonal functions and as such constitute the main ingredients in describing the spectral measures and spectral projections associated with the Laplacian on these spaces. In this note we strengthen this connection by showing that a set of spectral and geometric quantities associated with Jacobi operator fully describe the Maclaurin coefficients associated with the heat and other related Schwartzian kernels and present an explicit formulation of these quantities using the Bell polynomials.]]>