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Bessel integrals and fundamental solutions for a generalized Tricomi operator. (English) Zbl 0990.35004

Summary: The objective of the paper is the construction of a fundamental solution \(F\) for the generalized Tricomi operator \(y\Delta_x+\partial^2_y\), i.e., \((y\Delta_x+\partial^2_y)F=\delta_0(x)\otimes\delta_b(y)\). When \(b=0\), \(F\) is found explicitly by partial Fourier transformation with respect to the variables \(x=(x_1,\dots,x_n)\) (Theorem 4.1, Theorem 5.1). However, the authors “do not know how to obtain an explicit formula (or formulas) when \(b\neq 0\)” (p. 478). This explicit formula was established by A. Weinstein in two papers [Commun. Pure Appl. Math. 7, 105-116 (1954; Zbl 0056.09301), p. 105 (2.1), p. 106 (2.5), p. 107 (3.1, 3.2); Colloq. Int. CNRS 71, 179-186 (1956; Zbl 0075.08903), p. 181 (5.1)].
Explicit formulas for fundamental solutions of generalized Tricomi operators were also given in two papers of P. Germain and R. Bader [Rend. Circ. Mat. Palermo (2) 2, 53-70 (1953; Zbl 0052.09701); Bull. Soc. Math. Fr. 81, 145-174 (1953; Zbl 0051.07503)]. Germain and Bader also used the method of partial Fourier transformation with respect to \(x\) and expressed their result (also for \(b\neq 0\)) in terms of hypergeometric functions.
Explicit formulas for fundamental solutions of more general Euler-Poisson-Darboux and Tricomi-Clairaut operators were constructed in papers of S. Delache [Bull. Soc. Math. Fr. 97, 5-79 (1969; Zbl 0175.11003)] and S. Delache and J. Leray [Bull. Soc. Math. Fr. 99, 313-336 (1971; Zbl 0209.12701)]. In the last two papers the problem of defining distributions of the type \(T_\lambda=(1-|x|^2)+\lambda/\Gamma(\lambda+1)\), \(\lambda\in\mathbb C\), was solved correctly by understanding them as pull-backs of distributions by \(C^\infty\)-mappings. In the second paper of Weinstein cited above the fundamental solution was also represented as a Laplace transform of a product of Bessel functions (p. 182 (6.2)) – a connection which appears also in the paper under review (p. 482 (6.2), Lemma 3.2). There it serves to represent \(\mathcal F^{-1}(|\xi|^{-\nu}\mathcal I_\nu(|\xi|))\) as const \(T_{\nu-n/2}\) (Theorem 3.3). This result is given as an example of the computation of an \(n\)-dimensional Fourier transform in [L. Schwartz, Analyse IV: Applications à la théorie de la mesure (Hermann, Paris) (1993; Zbl 0920.00003), p. 162 (6.5.27)]. Laplace transforms of products of two Bessel functions are treated in the same manner as in the paper under review, in [T. C. Benton, SIAM J. Math. Anal. 6, No. 5, 761-765 (1975; Zbl 0314.33008)] and, more generally, in [B. C. Carlson, SIAM J. Math. Anal. 11, 428-435 (1980; Zbl 0467.44004)].
In Theorem 3.1, the (inverse) Fourier transform of \(|\xi|^\nu K_\nu(|\xi|)\) is evaluated. This is formula (VII, 10;13), p. 286, in [L. Schwartz, Théorie des distributions, Publ. Inst. Math. Univ. Strasbourg, IX–X. Nouvelle édition, entierement corrigée (Hermann, Paris) (1966; Zbl 0149.09501); Tome I, first edition (1950; Zbl 0037.07301); Tome II, first edition, (1951; Zbl 0042.11405)]. The (inverse) Fourier transforms of \(|\xi|^\nu\mathcal I_\nu(|\xi|)\) and \(|\xi|^\nu\mathcal N_\nu(|\xi|)\) in \(\mathbb R^n\) (Theorems 3.2, 3.4) seem to be new with the exception of the case \(\nu=0\) [cf. N. Kh. Ibragimov and E. V. Mamontov, Math. USSR, Sb. 31, 347-363 (1977; Zbl 0386.35027); translation from Mat. Sb., n. Ser. 102(144), 391-409 (1977; Zbl 0356.35055), Theorem 2.1].

MSC:

35A08 Fundamental solutions to PDEs
35M20 PDE of composite type (MSC2000)
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References:

[1] Barros-Neto, J., On Fundamental Solutions for the Tricomi Operator (1999), p. 69-88 · Zbl 0945.35063
[2] Barros-Neto, J.; Gelfand, I. M., Fundamental Solutions for the Tricomi Operator, Duke Math. J., 98, 465-483 (1999) · Zbl 0945.35063
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