On the hypoellipticity of convolution equations in the ultradistribution spaces of 𝐿^{𝑞} growth

Kovačević, Dušanka

doi:10.1090/S0002-9939-1994-1197540-4

On the hypoellipticity of convolution equations in the ultradistribution spaces of $L^ q$ growth
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Proc. Amer. Math. Soc. 120 (1994), 1181-1190 Request permission

Abstract:

We consider convolution equations in the ultradistribution spaces $\mathcal {D}_{{L^q}}^{’({M_p})}$ and $\mathcal {D}_{{L^q}}^{’\{ {M_p}\} },\;q \in [1,\infty ]$, of Beurling and Roumieu type of ${L^q}$ growth. Our main aim is to find conditions for convolution operators to be hypoelliptic in $\mathcal {D}_{{L^\infty }}^{’({M_p})}$ and $\mathcal {D}_{{L^\infty }}^{’\{ {M_p}\} }$ respectively, in terms of their Fourier transforms.

References

R. W. Braun, R. Meise, and D. Vogt, Existence of fundamental solutions and surjectivity of convolution operators on classes of ultra-differentiable functions, Proc. London Math. Soc. (3) 61 (1990), no. 2, 344–370. MR 1063049, DOI 10.1112/plms/s3-61.2.344
Richard D. Carmichael, R. S. Pathak, and S. Pilipović, Cauchy and Poisson integrals of ultradistributions, Complex Variables Theory Appl. 14 (1990), no. 1-4, 85–108. MR 1048709, DOI 10.1080/17476939008814408
Richard D. Carmichael, R. S. Pathak, and S. Pilipović, Holomorphic functions in tubes associated with ultradistributions, Complex Variables Theory Appl. 21 (1993), no. 1-2, 49–72. MR 1276560, DOI 10.1080/17476939308814614

La transformation de Fourier complexe et l’equation de convolution

Ioana Cioranescu, The characterization of the almost periodic ultradistributions of Beurling type, Proc. Amer. Math. Soc. 116 (1992), no. 1, 127–134. MR 1111214, DOI 10.1090/S0002-9939-1992-1111214-5

Convolution equations in

ultradistribution spaces

25

Konstruktion von fundamental lösungen für convolutoren in Roumieuschen ultradistributionsräumen

20

Hikosaburo Komatsu, Ultradistributions. I. Structure theorems and a characterization, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 20 (1973), 25–105. MR 320743

Ultradistributions

29

Some operations on the space

of tempered ultradistributions

Structural properties of the spaces of tempered ultradistributions

On kernels of slowly decreasing convolution operators

19

D. H. Pahk, On the convolution equations in the space of distributions of $L^p$-growth, Proc. Amer. Math. Soc. 94 (1985), no. 1, 81–86. MR 781061, DOI 10.1090/S0002-9939-1985-0781061-9

On the space

Stevan Pilipović, Characterizations of bounded sets in spaces of ultradistributions, Proc. Amer. Math. Soc. 120 (1994), no. 4, 1191–1206. MR 1211587, DOI 10.1090/S0002-9939-1994-1211587-0
Stevan Pilipović, Tempered ultradistributions, Boll. Un. Mat. Ital. B (7) 2 (1988), no. 2, 235–251 (English, with Italian summary). MR 946067

Multipliers, convolutors and hypoelliptic convolutors for tempered ultradistributions

Z. Zieleźny, Hypoelliptic and entire elliptic convolution equations in subspaces of the space of distributions. I, Studia Math. 28 (1966/67), 317–332. MR 222476, DOI 10.4064/sm-28-3-317-332
Z. Zieleźny, Hypoelliptic and entire elliptic convolution equations in subspaces of the space of distributions. II, Studia Math. 32 (1969), 47–59. MR 248521, DOI 10.4064/sm-32-1-47-59