A note on div curl inequalities. (English) Zbl 1113.26015
The authors obtain the following theorem: If \(u\) is a smooth \(q\)-form with compact support on \({\mathbb R}^n\), then:
\[ \begin{aligned} \text{(a)}\qquad &\| u\| _{L^{n/(n-1)}}\leq A[\| du\| _{L^1}+\| d^*u\| _{L^1}], \,\,\, \text{if} \,\,\, q\not=1,n-1;\\ \text{(b)}\qquad &\| u\| _{L^{n/(n-1)}}\leq A[\| du\| _{L^1}+\| d^*u\| _{H^1}], \,\,\, \text{if} \,\,\, q=1, \,\,\, \text{where}\,\,\, H^1\,\,\, \text{is the real Hardy space};\\ \text{(c)}\qquad &\| u\| _{L^{n/(n-1)}}\leq A[\| du\| _{H^1}+\| d^*u\| _{L^1}], \,\,\, \text{if} \,\,\, q=n-1.\end{aligned} \] These inequalities generalize the famous Gagliardo-Nirenberg inequality \((q=0)\) and a recent one \((q=1)\) obtained by J. Bourgain and H. Brezis [J. Am. Math. Soc. 16, No. 2, 393–426 (2003; Zbl 1075.35006)].
\[ \begin{aligned} \text{(a)}\qquad &\| u\| _{L^{n/(n-1)}}\leq A[\| du\| _{L^1}+\| d^*u\| _{L^1}], \,\,\, \text{if} \,\,\, q\not=1,n-1;\\ \text{(b)}\qquad &\| u\| _{L^{n/(n-1)}}\leq A[\| du\| _{L^1}+\| d^*u\| _{H^1}], \,\,\, \text{if} \,\,\, q=1, \,\,\, \text{where}\,\,\, H^1\,\,\, \text{is the real Hardy space};\\ \text{(c)}\qquad &\| u\| _{L^{n/(n-1)}}\leq A[\| du\| _{H^1}+\| d^*u\| _{L^1}], \,\,\, \text{if} \,\,\, q=n-1.\end{aligned} \] These inequalities generalize the famous Gagliardo-Nirenberg inequality \((q=0)\) and a recent one \((q=1)\) obtained by J. Bourgain and H. Brezis [J. Am. Math. Soc. 16, No. 2, 393–426 (2003; Zbl 1075.35006)].
Reviewer: Sotiris K. Ntouyas (Ioannina)
MSC:
| 26D10 | Inequalities involving derivatives and differential and integral operators |
| 58A10 | Differential forms in global analysis |
| 35B45 | A priori estimates in context of PDEs |
