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Saikat Kanjilal

Publications and source records attributed to Saikat Kanjilal.

5 recordsLinked to original sources
arXiv2026

Input-Endpoint Compactness and Essential Norms for Weighted Bilinear Hardy Operators

We study compactness of the product of two weighted Hardy primitives into $L^q(u)$, $0<q<\infty$, on the input boundary $1\le p_1,p_2\le\infty$, where at least one input exponent is $1$ or $\infty$. For $L^1\times L^p$, $1<p<\infty$, boundedness implies compactness when $q<p/(p+1)$. For $L^1\times L^1$, the corresponding range is $q<1/2$. At and above these boundaries compactness is characterized by additional active-tail decay, and critical power weights give bounded noncompact examples. The endpoint reductions use cumulative Stieltjes measures with their initial atoms and strict Hardy-Copson brackets. An output-mass allocation argument proves local finite-rank density, including quasi-Banach targets. Applying the localization principles developed for the interior-input regime identifies the essential distance, finite-rank distance and Hausdorff noncompactness radius with the coupled active tail, with the appropriate metric power when $q<1$. For a finite left endpoint, ordinary $L^\infty$ inputs admit an exact scalar transfer completed by linear nodal approximation.

math.FA↗
arXiv2026

Compactness and Essential Norms of Weighted Bilinear Hardy Operators on General Intervals

Let $I=(a,b)$ be an open interval with finite or infinite endpoints. For $1<p_1,p_2<\infty$ and $0<q<\infty$, we study the weighted bilinear Hardy operator $ \mathcal H_2(f,g)(x)=(\int_a^x f)(\int_a^x g) $ from $L^{p_1}(v_1;I)\times L^{p_2}(v_2;I)$ to $L^q(u;I)$, including the quasi-Banach target range $0<q<1$. With $\vartheta=\min\{1,q\}$, the target and bounded-bilinear-map spaces are treated through their complete powered metrics. Under explicit local integrability assumptions on the weights, compactness is equivalent to boundedness and vanishing endpoint-tail quasi-norms. Whenever the full operator is bounded, the $\vartheta$-power of the directed combined-tail limit equals exactly the powered essential distance, finite-rank distance, and Hausdorff measure of noncompactness in the target metric. Equivalently, the unpowered quantities agree under the stated quasi-norm-radius normalisation. Piecewise-constant middle-window approximants give quantitative bounds for the bilinear approximation numbers. The finite-rank-distance identity then implies that these numbers converge to the essential (quasi-)norm. Applying published norm-equivalent boundedness characteristics to endpoint-truncated output weights yields explicit compactness criteria in the convex, mixed, moderately subcritical, and deeply subcritical regimes. Algebraic companion exponents are used to make the negative powers arising for $q<1$ explicit. Four worked examples include an asymmetric mixed-range threshold and a nonlocally convex $L^{1/2}(0,1)$ example for which $a_{m+1}^{(2)}(\mathcal H_2)=O(m^{-1/2})$.

math.CA↗
arXiv2026

Endpoint-Safe Direct Characterisations of Weighted Bilinear Hardy Inequalities on Arbitrary Intervals

We study the weighted bilinear Hardy inequality for the same-direction product $H_I f H_I g$ on an arbitrary real interval $I$, with $0<q<\infty$, $1\le p_1,p_2\le\infty$, and measurable weights allowed to take the values $0$ and $+\infty$. The characterisation is formulated through directed compact restrictions $J\Subset I$. On each compact interval, endpoint-safe input profiles retain the $p_i=1$ boundary contribution as a Stieltjes atom, while the lower-triangle reduction separates a closed Hardy part from a strict Copson part so that common atoms are counted exactly once. Upper and mixed regimes follow from exact freezing and compact linear Hardy estimates. The lower regimes are obtained by power lifting and a measure-valued Hardy-Copson theorem. After simultaneous regularisation of the weights, the compact optimal constant is equivalent, with exponent-only constants, to an operational local characteristic, and the global constant satisfies $C_I\asymp\sup_{J\Subset I} A_J^{\mathrm{op}}$. In the regular interior range, the resulting conditions recover the classical $A_1,\ldots,A_7$ characterisations.

math.FA↗