Potential and Sobolev Spaces Related to Symmetrized Jacobi Expansions

Langowski, Bartosz

doi:10.3842/sigma.2015.073

Cited by 4 publications

(6 citation statements)

References 31 publications

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“…In some cases of orthogonal expansions, the theory was applied with emphasis on specific harmonic analysis issues. For instance, Langowski [7] studied the symmetrized Jacobi expansions with emphasis on potential and Sobolev spaces. See also [5].…”

Section: (): V-volmentioning

confidence: 99%

“…Sobolev spaces in the context of the Jacobi operator J α,β and the Jacobi-Dunkl operator J α,β , α, β > −1, were defined and investigated by Langowski [6,7]. These spaces were denoted W p,m α,β (for 1 p ∞ and order m 1) and their relation to potential spaces was studied.…”

Section: D-sobolev Spaces and The Friedrichs Extensionsmentioning

confidence: 99%

“…Consequently, also H k D,0 (I, w) is a Hilbert space. A comment is in order on relations between our Definition 4.1 and the definition of Sobolev spaces that appeared in [7] in the context of the Jacobi operator…”

Section: D-sobolev Spaces and The Friedrichs Extensionsmentioning

confidence: 99%

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Spectral Properties of Differential–Difference Symmetrized Operators

Stempak¹

2023

Mediterr. J. Math.

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We investigate some spectral properties of differential–difference operators, which are symmetrizations of differential operators of the form $$(\mathfrak {d}^\dagger \mathfrak {d})^k$$ ( d † d ) k and $$(\mathfrak {d}\mathfrak {d}^\dagger )^k$$ ( d d † ) k , $$k\geqslant 1$$ k ⩾ 1 . Here, $$\mathfrak {d}=p\frac{d}{\textrm{d}x} +q$$ d = p d d x + q and $$\mathfrak {d}^\dagger $$ d † stands for the formal adjoint of $$\mathfrak {d}$$ d on $$L^2((0,b),w\,\textrm{d}x)$$ L 2 ( ( 0 , b ) , w d x ) . In the simpliest case $$k=1$$ k = 1 , this symmetrization brings in the operator $$-\mathfrak {D}^2$$ - D 2 , which can be seen as a ‘Laplacian’, and $$\mathfrak {D}f:=\mathfrak {D}_{\mathfrak {d}}f= \mathfrak {d}(f_{\text {even}})-\mathfrak {d}^\dagger (f_{\text {odd}})$$ D f : = D d f = d ( f even ) - d † ( f odd ) , a skew-symmetric operator in $$L^2(I,w\,\textrm{d}x)$$ L 2 ( I , w d x ) , $$I=(-\,b,0)\cup (0,b)$$ I = ( - b , 0 ) ∪ ( 0 , b ) , is the symmetrization of $$\mathfrak {d}$$ d . Investigated spectral properties include self-adjoint extensions, among them the Friedrichs extensions, of the symmetrized operators.

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Section: (): V-volmentioning

confidence: 99%

Section: D-sobolev Spaces and The Friedrichs Extensionsmentioning

confidence: 99%

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Spectral Properties of Differential–Difference Symmetrized Operators

Stempak¹

2023

Mediterr. J. Math.

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“…According to our best knowledge, Sobolev spaces for Fourier-Bessel expansions were not investigated so far, in contrast to many other situations of classic orthogonal expansions, see e.g. [8,12,13,32,35,36,37].…”

Section: Introductionmentioning

confidence: 99%

On derivatives, Riesz transforms and Sobolev spaces for Fourier-Bessel expansions

Langowski¹,

Nowak²

2020

Preprint

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We study the problem of an appropriate choice of derivatives associated with discrete Fourier-Bessel expansions. We introduce a new so-called essential measure Fourier-Bessel setting, where the relevant derivative is simply the ordinary derivative. Then we investigate Riesz transforms and Sobolev spaces in this context. Our main results are L p -boundedness of the Riesz transforms (even in a multi-dimensional situation) and an isomorphism between the Sobolev and Fourier-Bessel potential spaces. Moreover, throughout the paper we collect various comments concerning two other closely related Fourier-Bessel situations that were considered earlier in the literature. We believe that our observations shed some new light on analysis of Fourier-Bessel expansions.

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“…Thus it is of interest and importance to look at the problem in concrete classical settings where proper tools and techniques are either known or can be effectively elaborated. Recently Langowski [22,24,25] verified that, in case of onedimensional Jacobi trigonometric polynomial and function contexts, the symmetrization leads to an extended setting admitting a good L p theory. In the present paper we take the opportunity to investigate another, this time multi-dimensional, concrete realization of the symmetrization procedure and find out that it admits a good L p theory as well, giving further support for the theory in [32].…”

Section: Introductionmentioning

confidence: 99%

On Harmonic Analysis Operators in Laguerre-Dunkl and Laguerre-Symmetrized Settings

Nowak¹,

Stempak²,

Szarek³

2016

SIGMA

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Abstract. We study several fundamental harmonic analysis operators in the multi-dimensional context of the Dunkl harmonic oscillator and the underlying group of reflections isomorphic to Z d 2 . Noteworthy, we admit negative values of the multiplicity functions. Our investigations include maximal operators, g-functions, Lusin area integrals, Riesz transforms and multipliers of Laplace and Laplace-Stieltjes type. By means of the general Calderón-Zygmund theory we prove that these operators are bounded on weighted L p spaces, 1 < p < ∞, and from weighted L 1 to weighted weak L 1 . We also obtain similar results for analogous set of operators in the closely related multi-dimensional Laguerre-symmetrized framework. The latter emerges from a symmetrization procedure proposed recently by the first two authors. As a by-product of the main developments we get some new results in the multi-dimensional Laguerre function setting of convolution type.

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Potential and Sobolev Spaces Related to Symmetrized Jacobi Expansions

Cited by 4 publications

References 31 publications

Spectral Properties of Differential–Difference Symmetrized Operators

Spectral Properties of Differential–Difference Symmetrized Operators

On derivatives, Riesz transforms and Sobolev spaces for Fourier-Bessel expansions

On Harmonic Analysis Operators in Laguerre-Dunkl and Laguerre-Symmetrized Settings

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