AlphaEvolve problems·alphaevolve:15

Gagliardo-Nirenberg Inequality

Let $1 \\leq q \\leq \\infty$, and let $j$ and $m$ be non-negative integers such that $j < m$. Furthermore, let $1 \\leq r \\leq \\infty, p \\geq 1$ be real and $\\theta \\in [0, 1]$ such that the following relations hold: $$\\frac{1}{p} = j + \\theta \\left( \\frac{1}{r} - m \\right) + \\frac{1 - \\theta}{q}, \\quad \\frac{j}{m} \\leq \\theta < 1.$$ Let $C(j,p,q,r,m)$ be the best constant such that $$\\|D^j u\\|_{L^p(\\mathbb{R})} \\leq C(j,p,q,r,m) \\|D^m u\\|_{L^r(\\mathbb{R})}^\\theta \\|u\\|_{L^q(\\mathbb{R})}^{1-\\theta}$$ for all test functions $u$, where $D$ denotes the derivative operator $\\frac{d}{dx}$. Establish upper and lower bounds for $C(j,p,q,r,m)$ that are as strong as possible.

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