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Transferring monotonicity in weighted norm inequalities.

Gord Sinnamon (2003)

Collectanea Mathematica

Certain weighted norm inequalities for integral operators with non-negative, monotone kernels are shown to remain valid when the weight is replaced by a monotone majorant or minorant of the original weight. A similar result holds for operators with quasi-concave kernels. To prove these results a careful investigation of the functional properties of the monotone envelopes of a non-negative function is carried-out. Applications are made to function space embeddings of the cones of monotone functions...

Two remarks about spectral asymptotics of pseudodifferential operators

Wojciech Czaja, Ziemowit Rzeszotnik (1999)

Colloquium Mathematicae

In this paper we show an asymptotic formula for the number of eigenvalues of a pseudodifferential operator. As a corollary we obtain a generalization of the result by Shubin and Tulovskiĭ about the Weyl asymptotic formula. We also consider a version of the Weyl formula for the quasi-classical asymptotics.

Two weight norm inequality for the fractional maximal operator and the fractional integral operator.

Yves Rakotondratsimba (1998)

Publicacions Matemàtiques

New sufficient conditions on the weight functions u(.) and v(.) are given in order that the fractional maximal [resp. integral] operator Ms [resp. Is], 0 ≤ s < n, [resp. 0 < s < n] sends the weighted Lebesgue space Lp(v(x)dx) into Lp(u(x)dx), 1 < p < ∞. As a consequence a characterization for this estimate is obtained whenever the weight functions are radial monotone.

Two-sided estimates for the approximation numbers of Hardy-type operators in L ∞ and L¹

W. Evans, D. Harris, J. Lang (1998)

Studia Mathematica

In [2] and [3] upper and lower estimates and asymptotic results were obtained for the approximation numbers of the operator T : L p ( ℝ + ) → L p ( ℝ + ) defined by ( T f ) ( x ) ≔ v ( x ) ʃ 0 ∞ u ( t ) f ( t ) d t when 1 < p < ∞. Analogous results are given in this paper for the cases p = 1,∞ not included in [2] and [3].

Two-sided estimates of the approximation numbers of certain Volterra integral operators

D. Edmunds, W. Evans, D. Harris (1997)

Studia Mathematica

We consider the Volterra integral operator T : L p ( ℝ + ) → L p ( ℝ + ) defined by ( T f ) ( x ) = v ( x ) ʃ 0 x u ( t ) f ( t ) d t . Under suitable conditions on u and v, upper and lower estimates for the approximation numbers a n ( T ) of T are established when 1 < p < ∞. When p = 2 these yield l i m n → ∞ n a n ( T ) = π - 1 ʃ 0 ∞ | u ( t ) v ( t ) | d t . We also provide upper and lower estimates for the ℓ α and weak ℓ α norms of (an(T)) when 1 < α < ∞.

Two-weighted criteria for integral transforms with multiple kernels

Vakhtang Kokilashvili, Alexander Meskhi (2006)

Banach Center Publications

Necessary and sufficient conditions governing two-weight L p norm estimates for multiple Hardy and potential operators are presented. Two-weight inequalities for potentials defined on nonhomogeneous spaces are also discussed. Sketches of the proofs for most of the results are given.

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И.Э. Вербицкий, Н.Я. Крупник (1980)

Matematiceskie issledovanija

Valuation of two-factor options under the Merton jump-diffusion model using orthogonal spline wavelets

Černá, Dana (2023)

Programs and Algorithms of Numerical Mathematics

This paper addresses the two-asset Merton model for option pricing represented by non-stationary integro-differential equations with two state variables. The drawback of most classical methods for solving these types of equations is that the matrices arising from discretization are full and ill-conditioned. In this paper, we first transform the equation using logarithmic prices, drift removal, and localization. Then, we apply the Galerkin method with a recently proposed orthogonal cubic spline-wavelet...

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