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On 1-regular ordinary differential operators

Grzegorz Łysik (2000)

Annales Polonici Mathematici

Solutions to singular linear ordinary differential equations with analytic coefficients are found in the form of Laplace type integrals.

On a testing-function space for distributions associated with the Kontorovich-Lebedev transform.

Semyon B. Yakubovich (2006)

Collectanea Mathematica

We construct a testing function space, which is equipped with the topology that is generated by Lν,p - multinorm of the differential operatorAx = x2 - x d/dx [x d/dx],and its k-th iterates Akx, where k = 0, 1, ... , and A0xφ = φ. Comparing with other testing-function spaces, we introduce in its dual the Kontorovich-Lebedev transformation for distributions with respect to a complex index. The existence, uniqueness, imbedding and inversion properties are investigated. As an application we find a solution...

On an integral transform by R. S. Phillips

Sten Bjon (2010)

Open Mathematics

The properties of a transformation f f ˜ h by R.S. Phillips, which transforms an exponentially bounded C 0-semigroup of operators T(t) to a Yosida approximation depending on h, are studied. The set of exponentially bounded, continuous functions f: [0, ∞[→ E with values in a sequentially complete L c-embedded space E is closed under the transformation. It is shown that ( f ˜ h ) k ˜ = f ˜ h + k for certain complex h and k, and that f ( t ) = lim h 0 + f ˜ h ( t ) , where the limit is uniform in t on compact subsets of the positive real line. If f is Hölder-continuous...

On bilinear Littlewood-Paley square functions.

Michael T. Lacey (1996)

Publicacions Matemàtiques

On the real line, let the Fourier transform of kn be k'n(ξ) = k'(ξ-n) where k'(ξ) is a smooth compactly supported function. Consider the bilinear operators Sn(f, g)(x) = ∫ f(x+y)g(x-y)kn(y) dy. If 2 ≤ p, q ≤ ∞, with 1/p + 1/q = 1/2, I prove thatΣ∞n=-∞ ||Sn(f,g)||22 ≤ C2||f||p2||g||q2.The constant C depends only upon k.

On conditions for the boundedness of the Weyl fractional integral on weighted L p spaces

Liliana De Rosa, Alberto de la Torre (2004)

Commentationes Mathematicae Universitatis Carolinae

In this paper we give a sufficient condition on the pair of weights ( w , v ) for the boundedness of the Weyl fractional integral I α + from L p ( v ) into L p ( w ) . Under some restrictions on w and v , this condition is also necessary. Besides, it allows us to show that for any p : 1 p < there exist non-trivial weights w such that I α + is bounded from L p ( w ) into itself, even in the case α > 1 .

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