On the estimation of the 𝐿₂-norm of a function over a bounded subset of 𝑅ⁿ

Walker, Homer F.

doi:10.1090/S0002-9939-1973-0312250-7

On the estimation of the $L_{2}$-norm of a function over a bounded subset of $\textbf {R}^{n}$
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by Homer F. Walker PDF

Proc. Amer. Math. Soc. 38 (1973), 103-110 Request permission

Abstract:

The objective of this paper is to present an estimate bounding the ${L_2}$-norm of a function over a bounded subset of ${R^n}$ by the ${L_2}$-norms of its derivatives of arbitrary order over all of ${R^n}$ and the ${L_2}$-norm of its projection onto a finite-dimensional space of functions with bounded support. The estimate essentially generalizes inequalities of Friedrichs [1, p. 284] and Lax and Phillips [2, p. 95]. An application of the estimate is made to the Fredholm theory of elliptic partial differential operators in ${R^n}$.

References

Avner Friedman, Partial differential equations of parabolic type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR 0181836
Peter D. Lax and Ralph S. Phillips, Scattering theory, Pure and Applied Mathematics, Vol. 26, Academic Press, New York-London, 1967. MR 0217440
Homer F. Walker, On the null-spaces of first-order elliptic partial differential operators in $R{\bf ^{n}}$, Proc. Amer. Math. Soc. 30 (1971), 278–286. MR 280864, DOI 10.1090/S0002-9939-1971-0280864-7
Homer F. Walker, A Fredholm theory for a class of first-order elliptic partial differential operators in $\textbf {R}^{n}$, Trans. Amer. Math. Soc. 165 (1972), 75–86. MR 291888, DOI 10.1090/S0002-9947-1972-0291888-3