File: Mathlib/Analysis/FunctionalSpaces/SobolevInequality.lean

full_name: «_aux_Mathlib_Analysis_FunctionalSpaces_SobolevInequality___macroRules_term#__1»
code: local prefix:max "#" => Fintype.card
goal: Lean.Macro

full_name: _aux_Mathlib_Analysis_FunctionalSpaces_SobolevInequality___unexpand_Fintype_card_1
code: local prefix:max "#" => Fintype.card
goal: Lean.PrettyPrinter.Unexpander

full_name: MeasureTheory.GridLines.T
code: /-- The "grid-lines operation" (not a standard name) which is central in the inductive proof of the
Sobolev inequality.

For a finite dependent product `Π i : ι, A i` of sigma-finite measure spaces, a finite set `s` of
indices from `ι`, and a (later assumed nonnegative) real number `p`, this operation acts on a
function `f` from `Π i, A i` into the extended nonnegative reals.  The operation is to partially
integrate, in the `s` co-ordinates, the function whose value at `x : Π i, A i` is obtained by
multiplying a certain power of `f` with the product, for each co-ordinate `i` in `s`, of a certain
power of the integral of `f` along the "grid line" in the `i` direction through `x`.

We are most interested in this operation when the set `s` is the universe in `ι`, but as a proxy for
"induction on dimension" we define it for a general set `s` of co-ordinates: the `s`-grid-lines
operation on a function `f` which is constant along the co-ordinates in `sᶜ` is morally (that is, up
to type-theoretic nonsense) the same thing as the universe-grid-lines operation on the associated
function on the "lower-dimensional" space `Π i : s, A i`. -/
def T (p : ℝ) (f : (∀ i, A i) → ℝ≥0∞) (s : Finset ι) : (∀ i, A i) → ℝ≥0∞ 
goal: ((i : ι) → A i) → ℝ≥0∞

full_name: MeasureTheory.GridLines.T_univ
code: @[simp] lemma T_univ (f : (∀ i, A i) → ℝ≥0∞) (x : ∀ i, A i) :
    T μ p f univ x =
    ∫⁻ (x : ∀ i, A i), (f x ^ (1 - (#ι - 1 : ℝ) * p)
    * ∏ i : ι, (∫⁻ t : A i, f (update x i t) ∂(μ i)) ^ p) ∂(.pi μ) 
goal: MeasureTheory.GridLines.T μ p f Finset.univ x =
  ∫⁻ (x : (i : ι) → A i),
    f x ^ (1 - (↑#ι - 1) * p) * ∏ i : ι, (∫⁻ (t : A i), f (Function.update x i t) ∂μ i) ^ p ∂MeasureTheory.Measure.pi μ

full_name: MeasureTheory.GridLines.T_empty
code: @[simp] lemma T_empty (f : (∀ i, A i) → ℝ≥0∞) (x : ∀ i, A i) :
    T μ p f ∅ x = f x ^ (1 + p) 
goal: MeasureTheory.GridLines.T μ p f ∅ x = f x ^ (1 + p)

full_name: MeasureTheory.GridLines.T_insert_le_T_lmarginal_singleton
code: /-- The main inductive step in the grid-lines lemma for the Gagliardo-Nirenberg-Sobolev inequality.

The grid-lines operation `GridLines.T` on a nonnegative function on a finitary product type is
less than or equal to the grid-lines operation of its partial integral in one co-ordinate
(the latter intuitively considered as a function on a space "one dimension down"). -/
theorem T_insert_le_T_lmarginal_singleton (hp₀ : 0 ≤ p) (s : Finset ι)
    (hp : (s.card : ℝ) * p ≤ 1)
    (i : ι) (hi : i ∉ s) {f : (∀ i, A i) → ℝ≥0∞} (hf : Measurable f) :
    T μ p f (insert i s) ≤ T μ p (∫⋯∫⁻_{i}, f ∂μ) s 
goal: MeasureTheory.GridLines.T μ p f (insert i s) ≤ MeasureTheory.GridLines.T μ p (∫⋯∫⁻_{i}, f ∂μ) s

full_name: MeasureTheory.GridLines.T_lmarginal_antitone
code: /-- Auxiliary result for the grid-lines lemma.  Given a nonnegative function on a finitary product
type indexed by `ι`, and a set `s` in `ι`, consider partially integrating over the variables in
`sᶜ` and performing the "grid-lines operation" (see `GridLines.T`) to the resulting function in the
variables `s`.  This theorem states that this operation decreases as the number of grid-lines taken
increases. -/
theorem T_lmarginal_antitone (hp₀ : 0 ≤ p) (hp : (#ι - 1 : ℝ) * p ≤ 1)
    {f : (∀ i, A i) → ℝ≥0∞} (hf : Measurable f) :
    Antitone (fun s ↦ T μ p (∫⋯∫⁻_sᶜ, f ∂μ) s) 
goal: Antitone fun s => MeasureTheory.GridLines.T μ p (∫⋯∫⁻_sᶜ, f ∂μ) s

full_name: MeasureTheory.lintegral_mul_prod_lintegral_pow_le
code: /-- The "grid-lines lemma" (not a standard name), stated with a general parameter `p` as the
exponent.  Compare with `lintegral_prod_lintegral_pow_le`.

For any finite dependent product `Π i : ι, A i` of sigma-finite measure spaces, for any
nonnegative real number `p` such that `(#ι - 1) * p ≤ 1`, for any function `f` from `Π i, A i` into
the extended nonnegative reals, we consider an associated "grid-lines quantity", the integral of an
associated function from `Π i, A i` into the extended nonnegative reals.  The value of this function
at `x : Π i, A i` is obtained by multiplying a certain power of `f` with the product, for each
co-ordinate `i`, of a certain power of the integral of `f` along the "grid line" in the `i`
direction through `x`.

This lemma bounds the Lebesgue integral of the grid-lines quantity by a power of the Lebesgue
integral of `f`. -/
theorem lintegral_mul_prod_lintegral_pow_le {p : ℝ} (hp₀ : 0 ≤ p)
    (hp : (#ι - 1 : ℝ) * p ≤ 1) {f : (∀ i : ι, A i) → ℝ≥0∞} (hf : Measurable f) :
    ∫⁻ x, f x ^ (1 - (#ι - 1 : ℝ) * p) * ∏ i, (∫⁻ xᵢ, f (update x i xᵢ) ∂μ i) ^ p ∂.pi μ
    ≤ (∫⁻ x, f x ∂.pi μ) ^ (1 + p) 
goal: ∫⁻ (x : (i : ι) → A i),
    f x ^ (1 - (↑#ι - 1) * p) *
      ∏ i : ι, (∫⁻ (xᵢ : A i), f (Function.update x i xᵢ) ∂μ i) ^ p ∂MeasureTheory.Measure.pi μ ≤
  (∫⁻ (x : (i : ι) → A i), f x ∂MeasureTheory.Measure.pi μ) ^ (1 + p)

full_name: MeasureTheory.lintegral_prod_lintegral_pow_le
code: /-- Special case of the grid-lines lemma `lintegral_mul_prod_lintegral_pow_le`, taking the extremal
exponent `p = (#ι - 1)⁻¹`. -/
theorem lintegral_prod_lintegral_pow_le
    {p : ℝ} (hp : Real.IsConjExponent #ι p)
    {f} (hf : Measurable f) :
    ∫⁻ x, ∏ i, (∫⁻ xᵢ, f (update x i xᵢ) ∂μ i) ^ ((1 : ℝ) / (#ι - 1 : ℝ)) ∂.pi μ
    ≤ (∫⁻ x, f x ∂.pi μ) ^ p 
goal: ∫⁻ (x : (i : ι) → A i),
    ∏ i : ι, (∫⁻ (xᵢ : A i), f (Function.update x i xᵢ) ∂μ i) ^ (1 / (↑#ι - 1)) ∂MeasureTheory.Measure.pi μ ≤
  (∫⁻ (x : (i : ι) → A i), f x ∂MeasureTheory.Measure.pi μ) ^ p

full_name: MeasureTheory.lintegral_pow_le_pow_lintegral_fderiv_aux
code: /-- The **Gagliardo-Nirenberg-Sobolev inequality**.  Let `u` be a continuously differentiable
compactly-supported function `u` on `ℝⁿ`, for `n ≥ 2`.  (More literally we encode `ℝⁿ` as
`ι → ℝ` where `n := #ι` is finite and at least 2.)  Then the Lebesgue integral of the pointwise
expression `|u x| ^ (n / (n - 1))` is bounded above by the `n / (n - 1)`-th power of the Lebesgue
integral of the Fréchet derivative of `u`.

For a basis-free version, see `lintegral_pow_le_pow_lintegral_fderiv`. -/
theorem lintegral_pow_le_pow_lintegral_fderiv_aux
    {p : ℝ} (hp : Real.IsConjExponent #ι p)
    {u : (ι → ℝ) → F} (hu : ContDiff ℝ 1 u)
    (h2u : HasCompactSupport u) :
    ∫⁻ x, (‖u x‖₊ : ℝ≥0∞) ^ p ≤ (∫⁻ x, ‖fderiv ℝ u x‖₊) ^ p 
goal: ∫⁻ (x : ι → ℝ), ↑‖u x‖₊ ^ p ≤ (∫⁻ (x : ι → ℝ), ↑‖fderiv ℝ u x‖₊) ^ p

full_name: MeasureTheory.wrapped._@.Mathlib.Analysis.FunctionalSpaces.SobolevInequality._hyg.5798
code: /-- The constant factor occurring in the conclusion of `lintegral_pow_le_pow_lintegral_fderiv`.
It only depends on `E`, `μ` and `p`.
It is determined by the ratio of the measures on `E` and `ℝⁿ` and
the operator norm of a chosen equivalence `E ≃ ℝⁿ` (raised to suitable powers involving `p`).-/
irreducible_def lintegralPowLePowLIntegralFDerivConst (p : ℝ) : ℝ≥0 
goal: Subtype (Eq @MeasureTheory.definition✝)

full_name: MeasureTheory.lintegralPowLePowLIntegralFDerivConst_def
code: /-- The constant factor occurring in the conclusion of `lintegral_pow_le_pow_lintegral_fderiv`.
It only depends on `E`, `μ` and `p`.
It is determined by the ratio of the measures on `E` and `ℝⁿ` and
the operator norm of a chosen equivalence `E ≃ ℝⁿ` (raised to suitable powers involving `p`).-/
irreducible_def lintegralPowLePowLIntegralFDerivConst (p : ℝ) : ℝ≥0 
goal: MeasureTheory.lintegralPowLePowLIntegralFDerivConst μ p =
  let ι := Fin (FiniteDimensional.finrank ℝ E);
  let_fun this := ⋯;
  let e := ContinuousLinearEquiv.ofFinrankEq this;
  let c := μ.addHaarScalarFactor (MeasureTheory.Measure.map (⇑e.symm) MeasureTheory.volume);
  c * ‖↑e.symm‖₊ ^ p * (c ^ p)⁻¹

full_name: MeasureTheory.lintegralPowLePowLIntegralFDerivConst
code: /-- The constant factor occurring in the conclusion of `lintegral_pow_le_pow_lintegral_fderiv`.
It only depends on `E`, `μ` and `p`.
It is determined by the ratio of the measures on `E` and `ℝⁿ` and
the operator norm of a chosen equivalence `E ≃ ℝⁿ` (raised to suitable powers involving `p`).-/
irreducible_def lintegralPowLePowLIntegralFDerivConst (p : ℝ) : ℝ≥0 
goal: ℝ≥0

full_name: MeasureTheory.definition._@.Mathlib.Analysis.FunctionalSpaces.SobolevInequality._hyg.5798
code: /-- The constant factor occurring in the conclusion of `lintegral_pow_le_pow_lintegral_fderiv`.
It only depends on `E`, `μ` and `p`.
It is determined by the ratio of the measures on `E` and `ℝⁿ` and
the operator norm of a chosen equivalence `E ≃ ℝⁿ` (raised to suitable powers involving `p`).-/
irreducible_def lintegralPowLePowLIntegralFDerivConst (p : ℝ) : ℝ≥0 
goal: ℝ≥0

full_name: MeasureTheory.lintegral_pow_le_pow_lintegral_fderiv
code: /-- The **Gagliardo-Nirenberg-Sobolev inequality**.  Let `u` be a continuously differentiable
compactly-supported function `u` on a normed space `E` of finite dimension `n ≥ 2`, equipped
with Haar measure. Then the Lebesgue integral of the pointwise expression
`|u x| ^ (n / (n - 1))` is bounded above by a constant times the `n / (n - 1)`-th power of the
Lebesgue integral of the Fréchet derivative of `u`. -/
theorem lintegral_pow_le_pow_lintegral_fderiv {u : E → F}
    (hu : ContDiff ℝ 1 u) (h2u : HasCompactSupport u)
    {p : ℝ} (hp : Real.IsConjExponent (finrank ℝ E) p) :
    ∫⁻ x, (‖u x‖₊ : ℝ≥0∞) ^ p ∂μ ≤
    lintegralPowLePowLIntegralFDerivConst μ p * (∫⁻ x, ‖fderiv ℝ u x‖₊ ∂μ) ^ p 
goal: ∫⁻ (x : E), ↑‖u x‖₊ ^ p ∂μ ≤
  ↑(MeasureTheory.lintegralPowLePowLIntegralFDerivConst μ p) * (∫⁻ (x : E), ↑‖fderiv ℝ u x‖₊ ∂μ) ^ p

full_name: MeasureTheory.wrapped._@.Mathlib.Analysis.FunctionalSpaces.SobolevInequality._hyg.8807
code: /-- The constant factor occurring in the conclusion of `eLpNorm_le_eLpNorm_fderiv_one`.
It only depends on `E`, `μ` and `p`. -/
irreducible_def eLpNormLESNormFDerivOneConst (p : ℝ) : ℝ≥0 
goal: Subtype (Eq @MeasureTheory.definition✝)

full_name: MeasureTheory.definition._@.Mathlib.Analysis.FunctionalSpaces.SobolevInequality._hyg.8807
code: /-- The constant factor occurring in the conclusion of `eLpNorm_le_eLpNorm_fderiv_one`.
It only depends on `E`, `μ` and `p`. -/
irreducible_def eLpNormLESNormFDerivOneConst (p : ℝ) : ℝ≥0 
goal: ℝ≥0

full_name: MeasureTheory.eLpNormLESNormFDerivOneConst_def
code: /-- The constant factor occurring in the conclusion of `eLpNorm_le_eLpNorm_fderiv_one`.
It only depends on `E`, `μ` and `p`. -/
irreducible_def eLpNormLESNormFDerivOneConst (p : ℝ) : ℝ≥0 
goal: MeasureTheory.eLpNormLESNormFDerivOneConst μ p = MeasureTheory.lintegralPowLePowLIntegralFDerivConst μ p ^ p⁻¹

full_name: MeasureTheory.eLpNormLESNormFDerivOneConst
code: /-- The constant factor occurring in the conclusion of `eLpNorm_le_eLpNorm_fderiv_one`.
It only depends on `E`, `μ` and `p`. -/
irreducible_def eLpNormLESNormFDerivOneConst (p : ℝ) : ℝ≥0 
goal: ℝ≥0

full_name: MeasureTheory.eLpNorm_le_eLpNorm_fderiv_one
code: /-- The **Gagliardo-Nirenberg-Sobolev inequality**.  Let `u` be a continuously differentiable
compactly-supported function `u` on a normed space `E` of finite dimension `n ≥ 2`, equipped
with Haar measure. Then the `Lᵖ` norm of `u`, where `p := n / (n - 1)`, is bounded above by
a constant times the `L¹` norm of the Fréchet derivative of `u`. -/
theorem eLpNorm_le_eLpNorm_fderiv_one  {u : E → F} (hu : ContDiff ℝ 1 u) (h2u : HasCompactSupport u)
    {p : ℝ≥0} (hp : NNReal.IsConjExponent (finrank ℝ E) p) :
    eLpNorm u p μ ≤ eLpNormLESNormFDerivOneConst μ p * eLpNorm (fderiv ℝ u) 1 μ 
goal: MeasureTheory.eLpNorm u (↑p) μ ≤
  ↑(MeasureTheory.eLpNormLESNormFDerivOneConst μ ↑p) * MeasureTheory.eLpNorm (fderiv ℝ u) 1 μ

full_name: MeasureTheory.eLpNormLESNormFDerivOfEqInnerConst
code: /-- The constant factor occurring in the conclusion of `eLpNorm_le_eLpNorm_fderiv_of_eq_inner`.
It only depends on `E`, `μ` and `p`. -/
def eLpNormLESNormFDerivOfEqInnerConst (p : ℝ) : ℝ≥0 
goal: ℝ≥0

full_name: MeasureTheory.eLpNorm_le_eLpNorm_fderiv_of_eq_inner
code: /-- The **Gagliardo-Nirenberg-Sobolev inequality**.  Let `u` be a continuously differentiable
compactly-supported function `u` on a normed space `E` of finite dimension `n`, equipped
with Haar measure, let `1 ≤ p < n` and let `p'⁻¹ := p⁻¹ - n⁻¹`.
Then the `Lᵖ'` norm of `u` is bounded above by a constant times the `Lᵖ` norm of
the Fréchet derivative of `u`.

Note: The codomain of `u` needs to be a Hilbert space.
-/
theorem eLpNorm_le_eLpNorm_fderiv_of_eq_inner  {u : E → F'}
    (hu : ContDiff ℝ 1 u) (h2u : HasCompactSupport u)
    {p p' : ℝ≥0} (hp : 1 ≤ p) (hn : 0 < finrank ℝ E)
    (hp' : (p' : ℝ)⁻¹ = p⁻¹ - (finrank ℝ E : ℝ)⁻¹) :
    eLpNorm u p' μ ≤ eLpNormLESNormFDerivOfEqInnerConst μ p * eLpNorm (fderiv ℝ u) p μ 
goal: MeasureTheory.eLpNorm u (↑p') μ ≤
  ↑(MeasureTheory.eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * MeasureTheory.eLpNorm (fderiv ℝ u) (↑p) μ

full_name: MeasureTheory.SNormLESNormFDerivOfEqConst_def
code: /-- The constant factor occurring in the conclusion of `eLpNorm_le_eLpNorm_fderiv_of_eq`.
It only depends on `E`, `F`, `μ` and `p`. -/
irreducible_def SNormLESNormFDerivOfEqConst [FiniteDimensional ℝ F] (p : ℝ) : ℝ≥0 
goal: MeasureTheory.SNormLESNormFDerivOfEqConst F μ p =
  let F' := EuclideanSpace ℝ (Fin (FiniteDimensional.finrank ℝ F));
  let e := toEuclidean;
  ‖↑e.symm‖₊ * MeasureTheory.eLpNormLESNormFDerivOfEqInnerConst μ p * ‖↑e‖₊

full_name: MeasureTheory.definition._@.Mathlib.Analysis.FunctionalSpaces.SobolevInequality._hyg.14110
code: /-- The constant factor occurring in the conclusion of `eLpNorm_le_eLpNorm_fderiv_of_eq`.
It only depends on `E`, `F`, `μ` and `p`. -/
irreducible_def SNormLESNormFDerivOfEqConst [FiniteDimensional ℝ F] (p : ℝ) : ℝ≥0 
goal: ℝ≥0

full_name: MeasureTheory.SNormLESNormFDerivOfEqConst
code: /-- The constant factor occurring in the conclusion of `eLpNorm_le_eLpNorm_fderiv_of_eq`.
It only depends on `E`, `F`, `μ` and `p`. -/
irreducible_def SNormLESNormFDerivOfEqConst [FiniteDimensional ℝ F] (p : ℝ) : ℝ≥0 
goal: ℝ≥0

full_name: MeasureTheory.wrapped._@.Mathlib.Analysis.FunctionalSpaces.SobolevInequality._hyg.14110
code: /-- The constant factor occurring in the conclusion of `eLpNorm_le_eLpNorm_fderiv_of_eq`.
It only depends on `E`, `F`, `μ` and `p`. -/
irreducible_def SNormLESNormFDerivOfEqConst [FiniteDimensional ℝ F] (p : ℝ) : ℝ≥0 
goal: Subtype (Eq MeasureTheory.definition✝)

full_name: MeasureTheory.eLpNorm_le_eLpNorm_fderiv_of_eq
code: /-- The **Gagliardo-Nirenberg-Sobolev inequality**.  Let `u` be a continuously differentiable
compactly-supported function `u` on a normed space `E` of finite dimension `n`, equipped
with Haar measure, let `1 < p < n` and let `p'⁻¹ := p⁻¹ - n⁻¹`.
Then the `Lᵖ'` norm of `u` is bounded above by a constant times the `Lᵖ` norm of
the Fréchet derivative of `u`.

This is the version where the codomain of `u` is a finite dimensional normed space.
-/
theorem eLpNorm_le_eLpNorm_fderiv_of_eq [FiniteDimensional ℝ F]
    {u : E → F} (hu : ContDiff ℝ 1 u) (h2u : HasCompactSupport u)
    {p p' : ℝ≥0} (hp : 1 ≤ p) (hn : 0 < finrank ℝ E)
    (hp' : (p' : ℝ)⁻¹ = p⁻¹ - (finrank ℝ E : ℝ)⁻¹) :
    eLpNorm u p' μ ≤ SNormLESNormFDerivOfEqConst F μ p * eLpNorm (fderiv ℝ u) p μ 
goal: MeasureTheory.eLpNorm u (↑p') μ ≤
  ↑(MeasureTheory.SNormLESNormFDerivOfEqConst F μ ↑p) * MeasureTheory.eLpNorm (fderiv ℝ u) (↑p) μ

full_name: MeasureTheory.wrapped._@.Mathlib.Analysis.FunctionalSpaces.SobolevInequality._hyg.15606
code: /-- The constant factor occurring in the conclusion of `eLpNorm_le_eLpNorm_fderiv_of_le`.
It only depends on `F`, `μ`, `s`, `p` and `q`. -/
irreducible_def eLpNormLESNormFDerivOfLeConst [FiniteDimensional ℝ F] (s : Set E) (p q : ℝ≥0) :
    ℝ≥0 
goal: Subtype (Eq MeasureTheory.definition✝)

full_name: MeasureTheory.eLpNormLESNormFDerivOfLeConst
code: /-- The constant factor occurring in the conclusion of `eLpNorm_le_eLpNorm_fderiv_of_le`.
It only depends on `F`, `μ`, `s`, `p` and `q`. -/
irreducible_def eLpNormLESNormFDerivOfLeConst [FiniteDimensional ℝ F] (s : Set E) (p q : ℝ≥0) :
    ℝ≥0 
goal: ℝ≥0

full_name: MeasureTheory.definition._@.Mathlib.Analysis.FunctionalSpaces.SobolevInequality._hyg.15606
code: /-- The constant factor occurring in the conclusion of `eLpNorm_le_eLpNorm_fderiv_of_le`.
It only depends on `F`, `μ`, `s`, `p` and `q`. -/
irreducible_def eLpNormLESNormFDerivOfLeConst [FiniteDimensional ℝ F] (s : Set E) (p q : ℝ≥0) :
    ℝ≥0 
goal: ℝ≥0

full_name: MeasureTheory.eLpNormLESNormFDerivOfLeConst_def
code: /-- The constant factor occurring in the conclusion of `eLpNorm_le_eLpNorm_fderiv_of_le`.
It only depends on `F`, `μ`, `s`, `p` and `q`. -/
irreducible_def eLpNormLESNormFDerivOfLeConst [FiniteDimensional ℝ F] (s : Set E) (p q : ℝ≥0) :
    ℝ≥0 
goal: MeasureTheory.eLpNormLESNormFDerivOfLeConst F μ s p q =
  let p' := (p⁻¹ - (↑(FiniteDimensional.finrank ℝ E))⁻¹)⁻¹;
  (μ s).toNNReal ^ (1 / ↑q - 1 / ↑p') * MeasureTheory.SNormLESNormFDerivOfEqConst F μ ↑p

full_name: MeasureTheory.eLpNorm_le_eLpNorm_fderiv_of_le
code: /-- The **Gagliardo-Nirenberg-Sobolev inequality**.  Let `u` be a continuously differentiable
function `u` supported in a bounded set `s` in a normed space `E` of finite dimension
`n`, equipped with Haar measure, and let `1 < p < n` and `0 < q ≤ (p⁻¹ - (finrank ℝ E : ℝ)⁻¹)⁻¹`.
Then the `L^q` norm of `u` is bounded above by a constant times the `Lᵖ` norm of
the Fréchet derivative of `u`.

Note: The codomain of `u` needs to be a finite dimensional normed space.
-/
theorem eLpNorm_le_eLpNorm_fderiv_of_le [FiniteDimensional ℝ F]
    {u : E → F} {s : Set E} (hu : ContDiff ℝ 1 u) (h2u : u.support ⊆ s)
    {p q : ℝ≥0} (hp : 1 ≤ p) (h2p : p < finrank ℝ E)
    (hpq : p⁻¹ - (finrank ℝ E : ℝ)⁻¹ ≤ (q : ℝ)⁻¹)
    (hs : Bornology.IsBounded s) :
    eLpNorm u q μ ≤ eLpNormLESNormFDerivOfLeConst F μ s p q * eLpNorm (fderiv ℝ u) p μ 
goal: MeasureTheory.eLpNorm u (↑q) μ ≤
  ↑(MeasureTheory.eLpNormLESNormFDerivOfLeConst F μ s p q) * MeasureTheory.eLpNorm (fderiv ℝ u) (↑p) μ

full_name: MeasureTheory.eLpNorm_le_eLpNorm_fderiv
code: /-- The **Gagliardo-Nirenberg-Sobolev inequality**.  Let `u` be a continuously differentiable
function `u` supported in a bounded set `s` in a normed space `E` of finite dimension
`n`, equipped with Haar measure, and let `1 < p < n`.
Then the `Lᵖ` norm of `u` is bounded above by a constant times the `Lᵖ` norm of
the Fréchet derivative of `u`.

Note: The codomain of `u` needs to be a finite dimensional normed space.
-/
theorem eLpNorm_le_eLpNorm_fderiv [FiniteDimensional ℝ F]
    {u : E → F} {s : Set E} (hu : ContDiff ℝ 1 u) (h2u : u.support ⊆ s)
    {p : ℝ≥0} (hp : 1 ≤ p) (h2p : p < finrank ℝ E) (hs : Bornology.IsBounded s) :
    eLpNorm u p μ ≤ eLpNormLESNormFDerivOfLeConst F μ s p p * eLpNorm (fderiv ℝ u) p μ 
goal: MeasureTheory.eLpNorm u (↑p) μ ≤
  ↑(MeasureTheory.eLpNormLESNormFDerivOfLeConst F μ s p p) * MeasureTheory.eLpNorm (fderiv ℝ u) (↑p) μ