We give a necessary and a sufficient condition under which the twodimensional Hardy operator is bounded between weighted Lehesgue spaces of functions of two variables with mixed norms. These conditions are expressed through boxindedness conditions on the one-dimensional Hardy operator between the corresponding weighted Lebesgue spaces of functions of one variable. AMS-Classification (1991): 46 E 30, 47 G 10 Let {X, and (Y, ||.||y) be two normed spaces of real functions over (0, A) and (0, B), respectively (0 < < oo). The space consists then, by definition, of all real functions / on (0,A) X {0,B) such that the function / (x , . ) belongs to Y for each X € (0, A), and the function x ||/(a:, .)l|y belongs to X. The linear space [X, Y] is usually equipped with the mixed norm The spaces [X,Y] have been studied in detail for Lebesgue spaces X and Y in [2]. They may be used, for instance, to characterize precisely those kernel functions which generate regulär integral operators between Lebesgue spaces (see e.g. [8,9]). In this note we shall be concerned with the case when X. and Y are weighted Lebesgue spaces. More precisely, we shall be interested in conditions under which the twodimensional Hardy operator (1) {H:,f){x,y) = m y { s , t ) d t d s 'This paper was written wbile the second author was visiting the University of Wützbutg. Financial suppoit of the Directorate General foi Science, Research and Development of the Commission of the European Coimnunities (Contract No. ERBCIPACT922095) is gratefully acknowledged. 92 Two-dimensional Hardy oper&toT is bounded from the space into the space [ I«(u) , I \" ( i ; ) ] , where U , V , u, and V are given weight functions, pi e (l,oo), and qt € (0,1) U (l ,oo) (t = 1,2). This amounts to proving the inequality (2) [joMJo^ {y)\\{H,f)ix,y)\\^dy

On the two-dimensional hardy operator in lebesgue spaces with mixed norms

The Duhamel product of functions f and g is defined by the formula 
$$(f \circledast g)(x) = \frac{d}{{dx}}\int\limits_0^x {f(x - t)g(t)dt.}$$

 . In the present paper, the Duhamel product is used in the study of spectral multiplicity for direct sums of operators and in the description of cyclic vectors of the restriction of the integration operator 
$$f(x,y) \mapsto \int\limits_0^x {\int\limits_0^y {f(t,\tau )d\tau \;dt} }$$

 in two variables to its invariant subspace consisting of functions that depend only on the product xy. Bibliography: 13 titles.

Some Applications of the Duhamel Product

Let C (n) A := C (n) A (D ◊ D) denote the subspace of functions in the Banach space C (n) D ◊ D which are analytic in the bi-disc D ◊ D. We con- sider the subspace Bzw consisting from the functions f 2 C (n) A which can be represented in the form f (z,w) = g (zw), where g is a single variable function from the disc algebra CA (D). We prove that Bzw is a Banach algebra under the Duhamel multiplication (f ~ g)(zw) = @ 2

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Maximal ideal space of some Banach algebras and related problems

Let 𝐶(𝑛)(𝔻×𝔻) be a Banach space of complex-valued functions 𝑓(𝑥,𝑦) that are continuous on 𝔻×𝔻, where 𝔻={𝑧∈ℂ∶|𝑧|<1} is the unit disc in the complex plane 
			ℂ, and have 	𝑛th partial derivatives in 𝔻×𝔻 which can be extended to functions continuous on 𝔻×𝔻, and let 𝐶𝐴(𝑛)=𝐶𝐴(𝑛)(𝔻×𝔻) denote the subspace of functions in 𝐶(𝑛)(𝔻×𝔻) which are analytic in 𝔻×𝔻 (i.e., 𝐶𝐴(𝑛)=𝐶(𝑛)(𝔻×𝔻)∩ℋ𝑜𝑙(𝔻×𝔻)). The double integration operator is defined in 𝐶𝐴(𝑛) by the formula ∫𝑊𝑓(𝑧,𝑤)=𝑧0∫𝑤0𝑓(𝑢,𝑣)𝑑𝑣𝑑𝑢. By using the method of Duhamel product for the functions in two variables, we describe the commutant of the restricted operator 𝑊∣𝐸𝑧𝑤, where 𝐸𝑧𝑤={𝑓∈𝐶𝐴(𝑛)∶𝑓(𝑧,𝑤)=𝑓(𝑧𝑤)} is an invariant subspace of 𝑊, and study its properties. We also study invertibility of the elements in 𝐶𝐴(𝑛) with respect to the Duhamel product.

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On Some Applications of a Special Integrodifferential Operators

Let C(n)=C(n)(D×D) be a Banach space of complex valued functions f(x,y) that are continuous on the closed bidisc D×D¯, where D={z∈C:|z|<1} is the unit disc in the complex plane C and has nth partial derivatives in D×D which can be extended to functions continuous on D×D¯. The Duhamel product is defined on C(n) by the formula (f⊛g)(z,w)=∂2∂z∂w∫0z∫0wf(z−u,w−v)g(u,v)dvdu. In the present paper we prove that C(n)(D×D) is a Banach algebra with respect to the Duhamel product ⊛. This result extends some known results. We also investigate the structure of the set of all extended eigenvalues and extended eigenvectors of some double integration operator Wzw. In particular, the commutant of the double integration operator Wzw is also described.

On the Banach algebra structure for C(n) of the bidisc and related topics

In this paper we consider the integration operator in two variables on L2[0, 1]2, determine its multiplicity and reducing subspaces, and make some observations about its invariant and hyperinvariant subspaces.

/pdf/the-integration-operator-in-two-variables-1qk78fbohj.pdf

A. Atzmon

Papers

The integration operator in two variables