Moser's inequality

The Moser inequalities are mathematical inequalities and are considered in the sub-area of functional analysis. They serve to estimate the norm of functions from the Sobolew spaces . They are named after the mathematician Jürgen Moser . They play an important role in the proof of existence of quasilinear systems , as the standardization is often used in these systems . ${\displaystyle L^{2}}$ ${\displaystyle L^{2}}$

Formulation of the Mos inequality

With is the space and with for the Sobolev space of the functions. Then there is a positive constant such that for all functions and for every multi-index with the inequality ${\displaystyle L^{p}(\mathbb {R} ^{m})}$ ${\displaystyle L^{p}}$ ${\displaystyle H^{s}(\mathbb {R} ^{m})}$ ${\displaystyle s\in \mathbb {N} }$ ${\displaystyle L^{2}}$ ${\displaystyle C_{s}\in \mathbb {R} _{>0}}$ ${\displaystyle f,\,g\in H^{s}(\mathbb {R} ^{m})\cap L^{\infty }(\mathbb {R} ^{m})}$ ${\displaystyle \alpha }$ ${\displaystyle \left|\alpha \right|=s}$

${\displaystyle \|D^{\alpha }(f\cdot g)\|_{L^{2}}\leqq C_{s}\left(\|f\|_{L^{\infty }}\|g\|_{L^{2}}+\|D^{s}f\|_{L^{2}}\|g\|_{L^{\infty }}\right)}$

applies.

If it is additionally assumed that one is weakly differentiable , that is, where the Sobolev space denotes the functions, then the inequality applies ${\displaystyle f}$ ${\displaystyle f\in H^{s}(\mathbb {R} ^{m})\cap W^{1,\infty }(\mathbb {R} ^{m})}$ ${\displaystyle W^{1,\infty }(\mathbb {R} ^{m})}$ ${\displaystyle L^{1}}$

${\displaystyle \|D^{\alpha }(fg)-fD^{\alpha }g\|_{L^{2}}\leqq C_{s}\left(\|D^{s}f\|_{L^{2}}\|g\|_{L^{\infty }}+\|f\|_{L^{\infty }}\|D^{s-1}g\|_{L^{2}}\right).}$

The function here is out of space and . ${\displaystyle g}$ ${\displaystyle H^{s-1}(\mathbb {R} ^{m})\cap L^{\infty }(\mathbb {R} ^{m})}$

These two inequalities are called Moser inequalities.

Proof idea

For the proof of the two inequalities one considers the special case first . Using the Leibniz rule , one then estimates the term with the Gagliardo-Nirenberg inequality . ${\displaystyle f,\,g\in C_{c}^{\infty }(\mathbb {R} ^{m})}$

Individual evidence

1. ^ ^{A b} Michael Eugene Taylor : Partial Differential Equations. Volume 3: Nonlinear equations. Springer, Berlin et al. 1996, ISBN 0-387-94652-7 ( Applied mathematical Sciences 117), pp. 10-11.